Jekyll2023-03-03T18:28:44+00:00https://sinhp.github.io/feed.xmlSina Hazratpourmy virtual dwellingSina Hazratpoursinahazratpour@gmail.comAbendlich strahlt der Sonne Auge2023-03-03T00:00:00+00:002023-03-03T00:00:00+00:00https://sinhp.github.io/posts/2023/03/abendlich_strahlt_der_Sonne_Auge<p>Conducted by Herbert von Karajan with Dietrich Fischer-Dieskau signing as Wotan (Berliner Philharmoniker).</p>
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<p><br />
<br /></p>
<p>Abendlich strahlt der Sonne Auge<br />
In prächtiger Glut<br />
Prangt glänzend die Burg<br />
In des Morgens Scheine mutig<br />
Erschimmernd, lag sie herrenlos<br />
Hehr verlockend vor mir<br /></p>
<p>Von Morgen bis Abend<br />
In Müh’ und Angst<br />
Nicht wonnig ward<br />
Sie gewonnen!<br /></p>
<p>Es naht die Nacht:<br />
Vor ihrem Neid<br />
Biete sie Bergung nun<br /></p>
<p>So grüss’ ich die Burg<br />
Sicher vor Bang’ und Grau’n!<br /></p>
<p>Folge mir, Frau:<br />
In Walhall<br />
Wohne mit mir!<br /></p>
<p><br />
<br /></p>
<p>In picture:</p>
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<p><br />
<br /></p>
<p>And here’s Hans Knappertsbutch’s version with Hans Hotter as Wotan. Bayreuth 1958.</p>
<iframe width="560" height="315" src="https://www.youtube.com/embed/EmDFiLwuovc" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen=""></iframe>SinaRheingold, RheingoldThe Forbidden Naiad of Wagner’s Music2023-01-05T00:00:00+00:002023-01-05T00:00:00+00:00https://sinhp.github.io/posts/2023/01/forbidden-naiad<p>Het Concert-Gebouw. Wagner’s die Walküre conducted by <a href="https://www.google.com/search?client=safari&rls=en&q=jaap+van+zwede+concert+gebouw&ie=UTF-8&oe=UTF-8">Jaap van Zwede</a>, Singers Stuart Skelton (Siegmund), Anja Kampe (Sieglinde), and bariton Kwangchul Youn (Hunding). It was a brilliant performance, way above my expectation.</p>
<p>The <a href="https://www.trouw.nl/cultuur-media/jaap-van-zweden-begint-met-een-briesje-maar-laat-het-dan-stormen-in-het-concertgebouw~bd9ea7f1/?referrer=https%3A%2F%2Fwww.google.com%2F">Trouw’s review</a> gets a badly wrong, in a way. It complains about the lack of staging, acting, forceful drama. It does not give enough credit to great singing of Anja Kampe (Sieglinde) and unblievable force in Skelton’s “Wälse! Wälse!” palpable at the bottom of my feet.</p>
<p>For me Wagner is not about drama, I don’t need any dramatic effect to be immersed in Wagner’s naiad. The music and singing is more than enough. Even Nietzche in his most anti-Wagnerian mode admitted this much.</p>
<blockquote>
<p>But quite apart from the magnétiseur and fresco-painter Wagner, there is another Wagner who lays aside small gems: our greatest melancholiac in music, full of glances, tendernesses, and comforting words in which nobody has anticipated him, the master in tones of a heavy-hearted and drowsy happiness.</p>
</blockquote>
<p>What is tenderness in Wagner, you ask. Listen to Knappertsbutsch’s performance of the Prelude to Act III of Wagner’s Meistersinger. The morning music for the one who has not slept all night.</p>
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<p>I think I have become one of the most corrupted Wagnerians.</p>Sinadie Walküre, conducted by Jaap van ZwedeRilke2022-12-01T00:00:00+00:002022-12-01T00:00:00+00:00https://sinhp.github.io/rilke<p>For one human being to love another; that is perhaps the most difficult of all our tasks, the ultimate, the last test and proof, the work for which all other work is but preparation. I hold this to be the highest task for a bond between two people: that each protects the solitude of the other.</p>Sina Hazratpoursinahazratpour@gmail.comFor one human being to love another; that is perhaps the most difficult of all our tasks, the ultimate, the last test and proof, the work for which all other work is but preparation. I hold this to be the highest task for a bond between two people: that each protects the solitude of the other.Mathematics and the mind2021-06-01T00:00:00+01:002021-06-01T00:00:00+01:00https://sinhp.github.io/posts/2021/06/cog-math<p><code class="language-plaintext highlighter-rouge">Slides:</code>
Some cognitive aspects of mathematics <a href="/files/Phil/philos_math/some_cognitive_aspects_of_math.pdf" target="_blank"> <i class="fa fa-file-pdf-o" aria-hidden="true"></i> </a></p>
<p><code class="language-plaintext highlighter-rouge">Video of the talk:</code>
<a href="https://youtu.be/MdsXEyRSIJk" target="_blank"> On YouTube </a></p>Sina HazratpourA review of cognitive accounts of mathematicsDie Sonne sinkt2021-03-21T00:00:00+00:002021-03-21T00:00:00+00:00https://sinhp.github.io/posts/2021/03/di-sonne-sinkt<p>Nicht lange durstest du noch, <br />
verbranntes Herz!<br />
Verheissung ist in der Luft,<br />
aus unbekannten Mündern bläst mich’s an<br />
— die grosse Kühle kommt <br /> <span> … <label for="sn-merging" class="margin-toggle sidenote-number"></label></span><input type="checkbox" id="sn-merging" class="margin-toggle" /><span class="sidenote"><em><a href="https://www.youtube.com/watch?v=zn0MWGHOb0s">hier</a> Dietrich Fischer-Dieskau singt das. </em></span></p>
<p>Meine Sonne stand heiss über mir im Mittage:<br />
seid mir gegrüsst, dass ihr kommt<br />
ihr plötzlichen Winde<br />
ihr kühlen Geister des Nachmittags!<br />
Die Luft geht fremd und rein.<br />
Schielt nicht mit schiefem<br />
Verführerblick<br />
die Nacht mich an? …<br />
Bleib stark, mein tapfres Herz!<br />
Frag nicht: warum? —<br /></p>
<p>Tag meines Lebens!<br />
die Sonne sinkt.<br />
Schon steht die glatte<br />
Fluth vergüldet.<br />
Warm athmet der Fels:<br />
schlief wohl zu Mittag<br />
das Glück auf ihm seinen Mittagsschlaf?<br />
In grünen Lichtern<br />
spielt Glück noch der braune Abgrund herauf.<br />
Tag meines Lebens!<br />
gen Abend gehts!<br />
Schon glüht dein Auge<br />
halbgebrochen,<br />
schon quillt deines Thaus<br />
Thränengeträufel,<br />
schon läuft still über weisse Meere<br />
deiner Liebe Purpur,<br />
deine letzte zögernde Seligkeit …<br /></p>
<p>Heiterkeit, güldene, komm!<br />
du des Todes<br />
heimlichster süssester Vorgenuss!<br />
— Lief ich zu rasch meines Wegs?<br />
Jetzt erst, wo der Fuss müde ward,<br />
holt dein Blick mich noch ein,<br />
holt dein Glück mich noch ein.<br />
Rings nur Welle und Spiel.<br />
Was je schwer war,<br />
sank in blaue Vergessenheit,<br />
müssig steht nun mein Kahn.<br />
Sturm und Fahrt — wie verlernt er das!<br />
Wunsch und Hoffen ertrank,<br />
glatt liegt Seele und Meer.<br /></p>
<p>Siebente Einsamkeit!<br />
Nie empfand ich<br />
näher mir süsse Sicherheit,<br />
wärmer der Sonne Blick.<br />
— Glüht nicht das Eis meiner Gipfel noch?<br />
Silbern, leicht, ein Fisch<br />
schwimmt nun mein Nachen hinaus …<br /></p>Friedrich NietzscheFriedrich Nietzsche, aus Dionysus-Dithyramben (1888)Mathematical Intuition According To Robert Musil2021-03-03T00:00:00+00:002021-03-03T00:00:00+00:00https://sinhp.github.io/posts/2021/03/intuition-and-the-man-without-qualities<p>Ulrich, the central protagonist of the multi-thousand page multi-volume novel <span>Der Mann ohne Eigenschaften<label for="sn-musil-der-mann" class="margin-toggle sidenote-number"></label></span><input type="checkbox" id="sn-musil-der-mann" class="margin-toggle" /><span class="sidenote"><em> The Man Without Qualities (German: Der Mann ohne Eigenschaften; 1930–1943) is an unfinished modernist novel in three volumes and various drafts, by the Austrian writer Robert Musil. Musil worked on the novel for more than twenty years. He started in 1921 and spent the rest of his life writing it. When he died in 1942, the novel was not completed. On Entitled Opinions you can listen to a <a href="https://entitledopinions.stanford.edu/hans-ulrich-gumbrecht-man-without-qualities">brilliant disuccsion</a> about the novel between Robbert Harrison and Hans Ulrich Gumbrecht. </em></span> by Robert Musil, has all qualities; in particular, he is a mathematician.</p>
<p>In <strong>“A chapter that may be skipped by anyone not particularly impressed by thinking as an occupation”</strong> we read about Ulrich’s inner thoughts as he remembers his recent conversation with Clarisse:</p>
<blockquote>
<p><em>Unfortunately, nothing is so hard to achieve as a literary representation of a man thinking. When someone asked a great scientist how he managed to come up with so much that was new, he replied: ‘‘Because I never stop thinking about it.” And it is surely safe to say that unexpected insights turn up for no other reason than that they are expected. They are in no small part a success of character, emotional stability, unflagging ambition, and unremitting work. What a bore such constancy must be. Looking at it another way, the solution of an intellectual problem comes about not very differently from a dog with a stick in his mouth trying to get through a narrow door; he will turn his head left and right until the stick slips through. We do much the same thing, but with the difference that we don’t make indiscriminate attempts but already know from experience approximately how it’s done. And if a clever fellow naturally has far more skill and experience with these twistings and turnings than a dim one, the slipping-through takes the clever fellow just as much by surprise; it is suddenly there, and one perceptibly feels slightly disconcerted because one’s ideas seem to have come of their own accord instead of waiting for their creator. This disconcerted feeling is nowadays called intuition by many people who would formerly, believing that it must be regarded as something suprapersonal, have called it inspiration; but it is only something impersonal, namely the affinity and coherence of the things themselves, meeting inside a head. The better the head, the less evident its presence in this process. As long as the process of thinking is in motion it is a quite wretched state, as if all the brain’s convolutions were suffering from colic; and when it is finished it no longer has the form of the thinking process as one experiences it but already that of what has been thought, which is regrettably impersonal, for the thought then faces outward and is dressed for communication to the world. When a man is in the process of thinking, there is no way to catch the moment between the personal and the impersonal, and this is manifestly why thinking is such an embarrassment for writers that they gladly avoid it.</em></p>
</blockquote>
<blockquote>
<p><em>But the man without qualities was now thinking.</em></p>
</blockquote>Sina HazratpourFrom The Man Without QualitiesTwo Dogmas of Empiricism by Willard Van Orman Quine2021-02-28T00:00:00+00:002021-02-28T00:00:00+00:00https://sinhp.github.io/posts/2021/02/two-dogmas-quine<p>Two Dogmas of Empiricism
Willard Van Orman Quine</p>
<p>From http://www.ditext.com/quine/quine.html</p>
<p>Originally published in The Philosophical Review 60 (1951): 20-43. Reprinted in W.V.O. Quine, From a Logical Point of View (Harvard University Press, 1953; second, revised, edition 1961), with the following alterations: “The version printed here diverges from the original in footnotes and in other minor respects: §§1 and 6 have been abridged where they encroach on the preceding essay [“On What There Is”], and §§3-4 have been expanded at points.”</p>
<p>Except for minor changes, additions and deletions are indicated in interspersed tables. I wish to thank Torstein Lindaas for bringing to my attention the need to distinguish more carefully the 1951 and the 1961 versions. Endnotes ending with an “a” are in the 1951 version; “b” in the 1961 version. (Andrew Chrucky, Feb. 15, 2000)</p>
<p>Modern empiricism has been conditioned in large part by two dogmas. One is a belief in some fundamental cleavage between truths which are analytic, or grounded in meanings independently of matters of fact and truths which are synthetic, or grounded in fact. The other dogma is reductionism: the belief that each meaningful statement is equivalent to some logical construct upon terms which refer to immediate experience. Both dogmas, I shall argue, are ill founded. One effect of abandoning them is, as we shall see, a blurring of the supposed boundary between speculative metaphysics and natural science. Another effect is a shift toward pragmatism.</p>
<ol>
<li>
<p>BACKGROUND FOR ANALYTICITY</p>
<p>Kant’s cleavage between analytic and synthetic truths was foreshadowed in Hume’s distinction between relations of ideas and matters of fact, and in Leibniz’s distinction between truths of reason and truths of fact. Leibniz spoke of the truths of reason as true in all possible worlds. Picturesqueness aside, this is to say that the truths of reason are those which could not possibly be false. In the same vein we hear analytic statements defined as statements whose denials are self-contradictory. But this definition has small explanatory value; for the notion of self-contradictoriness, in the quite broad sense needed for this definition of analyticity, stands in exactly the same need of clarification as does the notion of analyticity itself.2a The two notions are the two sides of a single dubious coin.</p>
<p>Kant conceived of an analytic statement as one that attributes to its subject no more than is already conceptually contained in the subject. This formulation has two shortcomings: it limits itself to statements of subject-predicate form, and it appeals to a notion of containment which is left at a metaphorical level. But Kant’s intent, evident more from the use he makes of the notion of analyticity than from his definition of it, can be restated thus: a statement is analytic when it is true by virtue of meanings and independently of fact. Pursuing this line, let us examine the concept of meaning which is presupposed.
(1951)
We must observe to begin with that meaning is not to be identified with naming or reference. Consider Frege’s example of ‘Evening Star’ and ‘Morning Star.’ Understood not merely as a recurrent evening apparition but as a body, the Evening Star is the planet Venus, and the Morning Star is the same. The two singular terms name the same thing. But the meanings must be treated as distinct, since the identity ‘Evening Star = Morning Star’ is a statement of fact established by astronomical observation. If ‘Evening Star’ and ‘Morning Star’ were alike in meaning, the identity ‘Evening Star = Morning Star’ would be analytic.
Again there is Russell’s example of ‘Scott’ and ‘the author of Waverly.’ Analysis of the meanings of words was by no means sufficient to reveal to George IV that the person named by these two singular terms was one and the same.</p>
<p>The distinction between meaning and naming is no less important at the level of abstract terms. The terms ‘9’ and ‘the number of planets’ name one and the same abstract entity but presumably must be regarded as unlike in meaning; for astronomical observation was needed, and not mere reflection on meanings, to determine the sameness of the entity in question.</p>
<p>Thus far we have been considering singular terms.</p>
</li>
</ol>
<p>(1961)
Meaning, let us remember, is not to be identified with naming.1b Frege’s example of ‘Evening Star’ and ‘Morning Star’ and Russell’s of ‘Scott’ and ‘the author of Waverly’, illustrate that terms can name the same thing but differ in meaning. The distinction between meaning and naming is no less important at the level of abstract terms. The terms ‘9’ and ‘the number of the planets’ name one and the same abstract entity but presumably must be regarded as unlike in meaning; for astronomical observation was needed, and not mere reflection on meanings, to determine the sameness of the entity in question.
The above examples consist of singular terms, concrete and abstract.</p>
<p>With general terms, or predicates, the situation is somewhat different but parallel. Whereas a singular term purports to name an entity, abstract or concrete, a general term does not; but a general term is true of an entity, or of each of many, or of none.2b The class of all entities of which a general term is true is called the extension of the term. Now paralleling the contrast between the meaning of a singular term and the entity named, we must distinguish equally between the meaning of a general term and its extension. The general terms ‘creature with a heart’ and ‘creature with a kidney,’ e.g., are perhaps alike in extension but unlike in meaning.</p>
<p>Confusion of meaning with extension, in the case of general terms, is less common than confusion of meaning with naming in the case of singular terms. It is indeed a commonplace in philosophy to oppose intension (or meaning) to extension, or, in a variant vocabulary, connotation to denotation.</p>
<p>The Aristotelian notion of essence was the forerunner, no doubt, of the modern notion of intension or meaning. For Aristotle it was essential in men to be rational, accidental to be two-legged. But there is an important difference between this attitude and the doctrine of meaning. From the latter point of view it may indeed be conceded (if only for the sake of argument) that rationality is involved in the meaning of the word ‘man’ while two-leggedness is not; but two-leggedness may at the same time be viewed as involved in the meaning of ‘biped’ while rationality is not. Thus from the point of view of the doctrine of meaning it makes no sense to say of the actual individual, who is at once a man and a biped, that his rationality is essential and his two-leggedness accidental or vice versa. Things had essences, for Aristotle, but only linguistic forms have meanings. Meaning is what essence becomes when it is divorced from the object of reference and wedded to the word.</p>
<p>For the theory of meaning the most conspicuous question is as to the nature of its objects: what sort of things are meanings?
(1951)
They are evidently intended to be ideas, somehow – mental ideas for some semanticists, Platonic ideas for others. Objects of either sort are so elusive, not to say debatable, that there seems little hope of erecting a fruitful science about them. It is not even clear, granted meanings, when we have two and when we have one; it is not clear when linguistic forms should be regarded as synonymous, or alike in meaning, and when they should not. If a standard of synonymy should be arrived at, we may reasonably expect that the appeal to meanings as entities will not have played a very useful part in the enterprise.
A felt need for meant entities may derive from an earlier failure to appreciate that meaning and reference are distinct. Once the theory of meaning is sharply separated from the theory of reference, it is a short step to recognizing as the business of the theory of meaning simply the synonymy of linguistic forms and the analyticity of statements; meanings themselves, as obscure intermediary entities, may well be abandoned.3b
(1951)
The description of analyticity as truth by virtue of meanings started us off in pursuit of a concept of meaning. But now we have abandoned the thought of any special realm of entities called meanings. So the problem of analyticity confronts us anew.
(1961)
The problem of analyticity confronts us anew.
Statements which are analytic by general philosophical acclaim are not, indeed, far to seek. They fall into two classes. Those of the first class, which may be called logically true, are typified by:</p>
<p>(1) No unmarried man is married.
The relevant feature of this example is that it is not merely true as it stands, but remains true under any and all reinterpretations of ‘man’ and ‘married.’ If we suppose a prior inventory of logical particles, comprising ‘no,’ ‘un-‘ ‘if,’ ‘then,’ ‘and,’ etc., then in general a logical truth is a statement which is true and remains true under all reinterpretations of its components other than the logical particles.
But there is also a second class of analytic statements, typified by:</p>
<p>(2) No bachelor is married.
The characteristic of such a statement is that it can be turned into a logical truth by putting synonyms for synonyms; thus (2) can be turned into (1) by putting ‘unmarried man’ for its synonym ‘bachelor.’ We still lack a proper characterization of this second class of analytic statements, and therewith of analyticity generally, inasmuch as we have had in the above description to lean on a notion of ‘synonymy’ which is no less in need of clarification than analyticity itself.
In recent years Carnap has tended to explain analyticity by appeal to what he calls state-descriptions.3a A state-description is any exhaustive assignment of truth values to the atomic, or noncompound, statements of the language. All other statements of the language are, Carnap assumes, built up of their component clauses by means of the familiar logical devices, in such a way that the truth value of any complex statement is fixed for each state-description by specifiable logical laws. A statement is then explained as analytic when it comes out true under every state-description. This account is an adaptation of Leibniz’s “true in all possible worlds.” But note that this version of analyticity serves its purpose only if the atomic statements of the language are, unlike ‘John is a bachelor’ and ‘John is married,’ mutually independent. Otherwise there would be a state-description which assigned truth to ‘John is a bachelor’ and falsity to ‘John is married,’ and consequently ‘All bachelors are married’ would turn out synthetic rather than analytic under the proposed criterion. Thus the criterion of analyticity in terms of state-descriptions serves only for languages devoid of extralogical synonym-pairs, such as ‘bachelor’ and ‘unmarried man’: synonym-pairs of the type which give rise to the “second class” of analytic statements. The criterion in terms of state-descriptions is a reconstruction at best of logical truth.</p>
<p>I do not mean to suggest that Carnap is under any illusions on this point. His simplified model language with its state-descriptions is aimed primarily not at the general problem of analyticity but at another purpose, the clarification of probability and induction. Our problem, however, is analyticity; and here the major difficulty lies not in the first class of analytic statements, the logical truths, but rather in the second class, which depends on the notion of synonymy.</p>
<p>II. DEFINITION
There are those who find it soothing to say that the analytic statements of the second class reduce to those of the first class, the logical truths, by definition; ‘bachelor,’ for example, is defined as ‘unmarried man.’ But how do we find that ‘bachelor’ is defined as ‘unmarried man’? Who defined it thus, and when? Are we to appeal to the nearest dictionary, and accept the lexicographer’s formulation as law? Clearly this would be to put the cart before the horse. The lexicographer is an empirical scientist, whose business is the recording of antecedent facts; and if he glosses ‘bachelor’ as ‘unmarried man’ it is because of his belief that there is a relation of synonymy between these forms, implicit in general or preferred usage prior to his own work. The notion of synonymy presupposed here has still to be clarified, presumably in terms relating to linguistic behavior. Certainly the “definition” which is the lexicographer’s report of an observed synonymy cannot be taken as the ground of the synonymy.</p>
<p>Definition is not, indeed, an activity exclusively of philologists. Philosophers and scientists frequently have occasions to “define” a recondite term by paraphrasing it into terms of a more familiar vocabulary. But ordinarily such a definition, like the philologist’s, is pure lexicography, affirming a relationship of synonymy antecedent to the exposition in hand.</p>
<p>Just what it means to affirm synonymy, just what the interconnections may be which are necessary and sufficient in order that two linguistic forms be properly describable as synonymous, is far from clear; but, whatever these interconnections may be, ordinarily they are grounded in usage. Definitions reporting selected instances of synonymy come then as reports upon usage.</p>
<p>There is also, however, a variant type of definitional activity which does not limit itself to the reporting of pre-existing synonymies. I have in mind what Carnap calls explication – an activity to which philosophers are given, and scientists also in their more philosophical moments. In explication the purpose is not merely to paraphrase the definiendum into an outright synonym, but actually to improve upon the definiendum by refining or supplementing its meaning. But even explication, though not merely reporting a pre-existing synonymy between definiendum and definiens, does rest nevertheless on other pre-existing synonymies. The matter may bc viewed as follows. Any word worth explicating has some contexts which, as wholes, are clear and precise enough to be useful; and the purpose of explication is to preserve the usage of these favored contexts while sharpening the usage of other contexts. In order that a given definition be suitable for purposes of explication, therefore, what is required is not that the definiendum in its antecedent usage be synonymous with the definiens, but just that each of these favored contexts of the definiendum taken as a whole in its antecedent usage, be synonymous with the corresponding context of the definiens.</p>
<p>Two alternative definientia may be equally appropriate for the purposes of a given task of explication and yet not be synonymous with each other; for they may serve interchangeably within the favored contexts but diverge elsewhere. By cleaving to one of these definientia rather than the other, a definition of explicative kind generates, by fiat, a relationship of synonymy between definiendum and definiens which did not hold before. But such a definition still owes its explicative function, as seen, to pre-existing synonymies.</p>
<p>There does, however, remain still an extreme sort of definition which does not hark back to prior synonymies at all; namely, the explicitly conventional introduction of novel notations for purposes of sheer abbreviation. Here the definiendum becomes synonymous with the definiens simply because it has been created expressly for the purpose of being synonymous with the definiens. Here we have a really transparent case of synonymy created by definition; would that all species of synonymy were as intelligible. For the rest, definition rests on synonymy rather than explaining it.</p>
<p>The word “definition” has come to have a dangerously reassuring sound, due no doubt to its frequent occurrence in logical and mathematical writings. We shall do well to digress now into a brief appraisal of the role of definition in formal work.</p>
<p>In logical and mathematical systems either of two mutually antagonistic types of economy may be striven for, and each has its peculiar practical utility. On the one hand we may seek economy of practical expression: ease and brevity in the statement of multifarious relationships. This sort of economy calls usually for distinctive concise notations for a wealth of concepts. Second, however, and oppositely, we may seek economy in grammar and vocabulary; we may try to find a minimum of basic concepts such that, once a distinctive notation has been appropriated to each of them, it becomes possible to express any desired further concept by mere combination and iteration of our basic notations. This second sort of economy is impractical in one way, since a poverty in basic idioms tends to a necessary lengthening of discourse. But it is practical in another way: it greatly simplifies theoretical discourse about the language, through minimizing the terms and the forms of construction wherein the language consists.</p>
<p>Both sorts of economy, though prima facie incompatible, are valuable in their separate ways. The custom has consequently arisen of combining both sorts of economy by forging in effect two languages, the one a part of the other. The inclusive language, though redundant in grammar and vocabulary, is economical in message lengths, while the part, called primitive notation, is economical in grammar and vocabulary. Whole and part are correlated by rules of translation whereby each idiom not in primitive notation is equated to some complex built up of primitive notation. These rules of translation are the so-called definitions which appear in formalized systems. They are best viewed not as adjuncts to one language but as correlations between two languages, the one a part of the other.</p>
<p>But these correlations are not arbitrary. They are supposed to show how the primitive notations can accomplish all purposes, save brevity and convenience, of the redundant language. Hence the definiendum and its definiens may be expected, in each case, to bc related in one or another of the three ways lately noted. The definiens may be a faithful paraphrase of the definiendum into the narrower notation, preserving a direct synonymy5b as of antecedent usage; or the definiens may, in the spirit of explication, improve upon the antecedent usage of the definiendum; or finally, the definiendum may be a newly created notation, newly endowed with meaning here and now.</p>
<p>In formal and informal work alike, thus, we find that definition – except in the extreme case of the explicitly conventional introduction of new notation – hinges on prior relationships of synonymy. Recognizing then that the notation of definition does not hold the key to synonymy and analyticity, let us look further into synonymy and say no more of definition.</p>
<p>III. INTERCHANGEABILITY
A natural suggestion, deserving close examination, is that the synonymy of two linguistic forms consists simply in their interchangeability in all contexts without change of truth value; interchangeability, in Leibniz’s phrase, salva veritate.5 6b Note that synonyms so conceived need not even be free from vagueness, as long as the vaguenesses match.</p>
<p>But it is not quite true that the synonyms ‘bachelor’ and ‘unmarried man’ are everywhere interchangeable salva veritate. Truths which become false under substitution of ‘unmarried man’ for ‘bachelor’ are easily constructed with help of ‘bachelor of arts’ or ‘bachelor’s buttons.’ Also with help of quotation, thus:</p>
<p>‘Bachelor’ has less than ten letters.
Such counterinstances can, however, perhaps be set aside by treating the phrases ‘bachelor of arts’ and ‘bachelor’s buttons’ and the quotation “bachelor” each as a single indivisible word and then stipulating that the interchangeability salva veritate which is to be the touchstone of synonymy is not supposed to apply to fragmentary occurrences inside of a word. This account of synonymy, supposing it acceptable on other counts, has indeed the drawback of appealing to a prior conception of “word” which can be counted on to present difficulties of formulation in its turn. Nevertheless some progress might be claimed in having reduced the problem of synonymy to a problem of wordhood. Let us pursue this line a bit, taking “word” for granted.
The question remains whether interchangeability salva veritate (apart from occurrences within words) is a strong enough condition for synonymy, or whether, on the contrary, some non-synonymous expressions might be thus interchangeable. Now let us be clear that we are not concerned here with synonymy in the sense of complete identity in psychological associations or poetic quality; indeed no two expressions are synonymous in such a sense. We are concerned only with what may be called cognitive synonymy. Just what this is cannot be said without successfully finishing the present study; but we know something about it from the need which arose for it in connection with analyticity in Section 1. The sort of synonymy needed there was merely such that any analytic statement could be turned into a logical truth by putting synonyms for synonyms. Turning the tables and assuming analyticity, indeed, we could explain cognitive synonymy of terms as follows (keeping to the familiar example): to say that ‘bachelor’ and ‘unmarried man’ are cognitively synonymous is to say no more nor less than that the statement:</p>
<p>(3) All and only bachelors are unmarried men
is analytic.4a 7b
What we need is an account of cognitive synonymy not presupposing analyticity – if we are to explain analyticity conversely with help of cognitive synonymy as undertaken in Section 1. And indeed such an independent account of cognitive synonymy is at present up for consideration, namely, interchangeability salva veritate everywhere except within words. The question before us, to resume the thread at last, is whether such interchangeability is a sufficient condition for cognitive synonymy. We can quickly assure ourselves that it is, by examples of the following sort. The statement:</p>
<p>(4) Necessarily all and only bachelors are bachelors
is evidently true, even supposing ‘necessarily’ so narrowly construed as to be truly applicable only to analytic statements. Then, if ‘bachelor’ and ‘unmarried man’ are interchangeable salva veritate, the result
(5) Necessarily, all and only bachelors are unmarried men
of putting ‘unmarried man’ for an occurrence of ‘bachelor’ in (4) must, like (4), be true. But to say that (5) is true is to say that (3) is analytic, and hence that ‘bachelor’ and ‘unmarried man’ are cognitively synonymous.
Let us see what there is about the above argument that gives it its air of hocus-pocus. The condition of interchangeability salva veritate varies in its force with variations in the richness of the language at hand. The above argument supposes we are working with a language rich enough to contain the adverb ‘necessarily,’ this adverb being so construed as to yield truth when and only when applied to an analytic statement. But can we condone a language which contains such an adverb? Does the adverb really make sense? To suppose that it does is to suppose that we have already made satisfactory sense of ‘analytic.’ Then what are we so hard at work on right now?</p>
<p>Our argument is not flatly circular, but something like it. It has the form, figuratively speaking, of a closed curve in space.</p>
<p>Interchangeability salva veritate is meaningless until relativized to a language whose extent is specified in relevant respects. Suppose now we consider a language containing just the following materials. There is an indefinitely large stock of one- and many-place predicates,
(1951)
There is an indefinitely large stock of one- and many-place predicates,
(1961)
There is an indefinitely large stock of one-place predicates, (for example, ‘F’ where ‘Fx’ means that x is a man) and many-placed predicates (for example, ‘G’ where ‘Gxy’ means that x loves y,
mostly having to do with extralogical subject matter. The rest of the language is logical. The atomic sentences consist each of a predicate followed by one or more variables ‘x’, ‘y’, etc.; and the complex sentences are built up of atomic ones by truth functions (‘not’, ‘and’, ‘or’, etc.) and quantification.8b In effect such a language enjoys the benefits also of descriptions and class names and indeed singular terms generally, these being contextually definable in known ways.5a 9b</p>
<p>(1961)
Even abstract singular terms naming classes, classes of classes, etc., are contextually definable in case the assumed stock of predicates includes the two-place predicate of class membership.10b
(1951)
Such a language can be adequate to classical mathematics and indeed to scientific discourse generally, except in so far as the latter involves debatable devices such as modal adverbs and contrary-to-fact conditionals.
(1961)
Such a language can be adequate to classical mathematics and indeed to scientific discourse generally, except in so far as the latter involves debatable devices such as contrary-to-fact conditionals or modal adverbs like ‘necessarily’.11b
Now a language of this type is extensional, in this sense: any two predicates which agree extensionally (i.e., are true of the same objects) are interchangeable salva veritate.12b</p>
<p>In an extensional language, therefore, interchangeability salva veritate is no assurance of cognitive synonymy of the desired type. That ‘bachelor’ and ‘unmarried man’ are interchangeable salva veritate in an extensional language assures us of no more than that (3) is true. There is no assurance here that the extensional agreement of ‘bachelor’ and ‘unmarried man’ rests on meaning rather than merely on accidental matters of fact, as does extensional agreement of ‘creature with a heart’ and ‘creature with a kidney.’</p>
<p>For most purposes extensional agreement is the nearest approximation to synonymy we need care about. But the fact remains that extensional agreement falls far short of cognitive synonymy of the type required for explaining analyticity in the manner of Section I. The type of cognitive synonymy required there is such as to equate the synonymy of ‘bachelor’ and ‘unmarried man’ with the analyticity of (3), not merely with the truth of (3).</p>
<p>So we must recognize that interchangeability salva veritate, if construed in relation to an extensional language, is not a sufficient condition of cognitive synonymy in the sense needed for deriving analyticity in the manner of Section I. If a language contains an intensional adverb ‘necessarily’ in the sense lately noted, or other particles to the same effect, then interchangeability salva veritate in such a language does afford a sufficient condition of cognitive synonymy; but such a language is intelligible only if the notion of analyticity is already clearly understood in advance.</p>
<p>The effort to explain cognitive synonymy first, for the sake of deriving analyticity from it afterward as in Section I, is perhaps the wrong approach. Instead we might try explaining analyticity somehow without appeal to cognitive synonymy. Afterward we could doubtless derive cognitive synonymy from analyticity satisfactorily enough if desired. We have seen that cognitive synonymy of ‘bachelor’ and ‘unmarried man’ can be explained as analyticity of (3). The same explanation works for any pair of one-place predicates, of course, and it can be extended in obvious fashion to many-place predicates. Other syntactical categories can also he accommodated in fairly parallel fashion. Singular terms may be said to be cognitively synonymous when the statement of identity formed by putting ‘=’ between them is analytic. Statements may be said simply to be cognitively synonymous when their biconditional (the result of joining them by ‘if and only if’) is analytic.6a 13b If we care to lump all categories into a single formulation, at the expense of assuming again the notion of “word” which was appealed to early in this section, we can describe any two linguistic forms as cognitively synonymous when the two forms are interchangeable (apart from occurrences within “words”) salva (no longer veritate but) analyticitate. Certain technical questions arise, indeed, over cases of ambiguity or homonymy; let us not pause for them, however, for we are already digressing. Let us rather turn our backs on the problem of synonymy and address ourselves anew to that of analyticity.</p>
<p>IV. SEMANTICAL RULES
Analyticity at first seemed most naturally definable by appeal to a realm of meanings. On refinement, the appeal to meanings gave way to an appeal to synonymy or definition. But definition turned out to be a will-o’-the-wisp, and synonymy turned out to be best understood only by dint of a prior appeal to analyticity itself. So we are back at the problem of analyticity.</p>
<p>I do not know whether the statement ‘Everything green is extended’ is analytic. Now does my indecision over this example really betray an incomplete understanding, an incomplete grasp of the “meanings,” of ‘green’ and ‘extended’? I think not. The trouble is not with ‘green’ or ‘extended,’ but with ‘analytic.’</p>
<p>It is often hinted that the difficulty in separating analytic statements from synthetic ones in ordinary language is due to the vagueness of ordinary language and that the distinction is clear when we have a precise artificial language with explicit “semantical rules.” This, however, as I shall now attempt to show, is a confusion.</p>
<p>The notion of analyticity about which we are worrying is a purported relation between statements and languages: a statement S is said to be analytic for a language L, and the problem is to make sense of this relation generally, for example, for variable ‘S’ and ‘L.’ The point that I want to make is that the gravity of this problem is not perceptibly less for artificial languages than for natural ones. The problem of making sense of the idiom ‘S is analytic for L,’ with variable ‘S’ and ‘L,’ retains its stubbornness even if we limit the range of the variable ‘L’ to artificial languages. Let me now try to make this point evident.</p>
<p>For artificial languages and semantical rules we look naturally to the writings of Carnap. His semantical rules take various forms, and to make my point I shall have to distinguish certain of the forms. Let us suppose, to begin with, an artificial language L0 whose semantical rules have the form explicitly of a specification, by recursion or otherwise, of all the analytic statements of L0. The rules tell us that such and such statements, and only those, are the analytic statements of L0. Now here the difficulty is simply that the rules contain the word ‘analytic,’ which we do not understand! We understand what expressions the rules attribute analyticity to, but we do not understand what the rules attribute to those expressions. In short, before we can understand a rule which begins “A statement S is analytic for language L0 if and only if . . . ,” we must understand the general relative term ‘analytic for’; we must understand ‘S is analytic for L’ where ‘S’ and ‘L’ are variables.</p>
<p>Alternatively we may, indeed, view the so-called rule as a conventional definition of a new simple symbol ‘analytic-for-L0,’ which might better be written untendentiously as ‘K’ so as not to seem to throw light on the interesting word “analytic.” Obviously any number of classes K, M, N, etc., of statements of L0 can be specified for various purposes or for no purpose; what does it mean to say that K, as against M, N, etc., is the class of the ‘analytic’ statements of L0?</p>
<p>By saying what statements are analytic for L0 we explain ‘analytic-for L0 ‘ but not ‘analytic for.’ We do not begin to explain the idiom ‘S is analytic for L’ with variable ‘S’ and ‘L,’ even though we be content to limit the range of ‘L’ to the realm of artificial languages.</p>
<p>Actually we do know enough about the intended significance of ‘analytic’ to know that analytic statements are supposed to be true. Let us then turn to a second form of semantical rule, which says not that such and such statements are analytic but simply that such and such statements are included among the truths. Such a rule is not subject to the criticism of containing the un-understood word ‘analytic’; and we may grant for the sake of argument that there is no difficulty over the broader term ‘true.’ A semantical rule of this second type, a rule of truth, is not supposed to specify all the truths of the language; it merely stipulates, recursively or otherwise, a certain multitude of statements which, along with others unspecified, are to count as true. Such a rule may be conceded to be quite clear. Derivatively, afterward, analyticity can be demarcated thus: a statement is analytic if it is (not merely true but) true according to the semantical rule.</p>
<p>Still there is really no progress. Instead of appealing to an unexplained word ‘analytic,’ we are now appealing to an unexplained phrase ‘semantical rule.’ Not every true statement which says that the statements of some class are true can count as a semantical rule – otherwise all truths would be “analytic” in the sense of being true according to semantical rules. Semantical rules are distinguishable, apparently, only by the fact of appearing on a page under the heading ‘Semantical Rules’; and this heading is itself then meaningless.</p>
<p>We can say indeed that a statement is analytic-for-L0 if and only if it is true according to such and such specifically appended “semantical rules,” but then we find ourselves back at essentially the same case which was originally discussed: ‘S is analytic-for-L0 if and only if. . . .’ Once we seek to explain ‘S is analytic for L’ generally for variable ‘L’ ( even allowing limitation of ‘L’ to artificial languages ), the explanation ‘true according to the semantical rules of L’ is unavailing; for the relative term ‘semantical rule of’ is as much in need of clarification, at least, as ‘analytic for.’</p>
<p>(1961)
It may be instructive to compare the notion of semantical rule with that of postulate. Relative to the given set of postulates, it is easy to say that what a postulate is: it is a member of the set. Relative to a given set of semantical rules, it is equally easy to say what a semantical rule is. But given simply a notation, mathematical or otherwise, and indeed as thoroughly understood a notation as you please in point of the translation or truth conditions of its statements, who can say which of its true statements rank as postulates? Obviously the question is meaningless – as meaningless as asking which points in Ohio are starting points. Any finite (or effectively specifiable infinite) selection of statements (preferably true ones, perhaps) is as much a set of postulates as any other. The word ‘postulate’ is significant only relative to an act of inquiry; we apply the word to a set of statements just in so far as we happen, for the year or the argument, to be thinking of those statements which can be reached from them by some set of trasformations to which we have seen fit to direct our attention. Now the notion of semantical rule is as sensible and meaningful as that of postulate, if conceived in a similarly relative spirit – relative, this time, to one or another particular enterprise of schooling unconversant persons in sufficient conditions for truth of statements of some natural or artificial language L. But from this point of view no one signalization of a subclass of the truths of L is intrinsically more a semantical rule than another; and, if ‘analytic’ means ‘true by semantical rules’, no one truth of L is analytic to the exclusion of another. 14b
It might conceivably be protested that an artificial language L (unlike a natural one) is a language in the ordinary sense plus a set of explicit semantical rules – the whole constituting, let us say, an ordered pair; and that the semantical rules of L then are specifiable simply as the second component of the pair L. But, by the same token and more simply, we might construe an artificial language L outright as an ordered pair whose second component is the class of its analytic statements; and then the analytic statements of L become specifiable simply as the statements in the second component of L. Or better still, we might just stop tugging at our bootstraps altogether.</p>
<p>Not all the explanations of analyticity known to Carnap and his readers have been covered explicitly in the above considerations, but the extension to other forms is not hard to see. Just one additional factor should be mentioned which sometimes enters: sometimes the semantical rules are in effect rules of translation into ordinary language, in which case the analytic statements of the artificial language are in effect recognized as such from the analyticity of their specified translations in ordinary language. Here certainly there can be no thought of an illumination of the problem of analyticity from the side of the artificial language.</p>
<p>From the point of view of the problem of analyticity the notion of an artificial language with semantical rules is a feu follet par ercellence. Semantical rules determining the analytic statements of an artificial language are of interest only in so far as we already understand the notion of analyticity; they are of no help in gaining this understanding.</p>
<p>Appeal to hypothetical languages of an artificially simple kind could conceivably bc useful in clarifying analyticity, if the mental or behavioral or cultural factors relevant to analyticity – whatever they may be – were somehow sketched into the simplified model. But a model which takes analyticity merely as an irreducible character is unlikely to throw light on the problem of explicating analyticity.</p>
<p>It is obvious that truth in general depends on both language and extra-linguistic fact. The statement ‘Brutus killed Caesar’ would be false if the world had been different in certain ways, but it would also be false if the word ‘killed’ happened rather to have the sense of ‘begat.’ Hence the temptation to suppose in general that the truth of a statement is somehow analyzable into a linguistic component and a factual component. Given this supposition, it next seems reasonable that in some statements the factual component should be null; and these are the analytic statements. But, for all its a priori reasonableness, a boundary between analytic and synthetic statement simply has not been drawn. That there is such a distinction to be drawn at all is an unempirical dogma of empiricists, a metaphysical article of faith.</p>
<p>V. THE VERIFICATION THEORY AND REDUCTIONISM
In the course of these somber reflections we have taken a dim view first of the notion of meaning, then of the notion of cognitive synonymy: and finally of the notion of analyticity. But what, it may be asked, of the verification theory of meaning? This phrase has established itself so firmly as a catchword of empiricism that we should be very unscientific indeed not to look beneath it for a possible key to the problem of meaning and the associated problems.</p>
<p>The verification theory of meaning, which has been conspicuous in the literature from Peirce onward, is that the meaning of a statement is the method of empirically confirming or infirming it. An analytic statement is that limiting case which is confirmed no matter what.</p>
<p>As urged in Section I, we can as well pass over the question of meanings as entities and move straight to sameness of meaning, or synonymy. Then what the verification theory says is that statements are synonymous if and only if they are alike in point of method of empirical confirmation or infirmation.</p>
<p>This is an account of cognitive synonymy not of linguistic forms generally, but of statements.7a 15b However, from the concept of synonymy of statements we could derive the concept of synonymy for other linguistic forms, by considerations somewhat similar to those at the end of Section III. Assuming the notion of “word,” indeed, we could explain any two forms as synonymous when the putting of the one form for an occurrence of the other in any statement (apart from occurrences within “words”) yields a synonymous statement. Finally, given the concept of synonymy thus for linguistic forms generally, we could define analyticity in terms of synonymy and logical truth as in Section I. For that matter, we could define analyticity more simply in terms of just synonymy of statements together with logical truth; it is not necessary to appeal to synonymy of linguistic forms other than statements. For a statement may be described as analytic simply when it is synonymous with a logically true statement.</p>
<p>So, if the verification theory can be accepted as an adequate account of statement synonymy, the notion of analyticity is saved after all. However, let us reflect. Statement synonymy is said to be likeness of method of empirical confirmation or infirmation. Just what are these methods which are to be compared for likeness? What, in other words, is the nature of the relationship between a statement and the experiences which contribute to or detract from its confirmation?</p>
<p>The most naive view of the relationship is that it is one of direct report. This is radical reductionism. Every meaningful statement is held to be translatable into a statement (true or false) about immediate experience. Radical reductionism, in one form or another, well antedates the verification theory of meaning explicitly so called. Thus Locke and Hume held that every idea must either originate directly in sense experience or else be compounded of ideas thus originating; and taking a hint from Tooke7a we might rephrase this doctrine in semantical jargon by saying that a term, to be significant at all, must be either a name of a sense datum or a compound of such names or an abbreviation of such a compound. So stated, the doctrine remains ambiguous as between sense data as sensory events and sense data as sensory qualities; and it remains vague as to the admissible ways of compounding. Moreover, the doctrine is unnecessarily and intolerably restrictive in the term-by-term critique which it imposes. More reasonably, and without yet exceeding the limits of what I have called radical reductionism, we may take full statements as our significant units – thus demanding that our statements as wholes be translatable into sense-datum language, but not that they be translatable term by term.
(1951)
This emendation would unquestionably have been welcome to Locke and Hume and Tooke, but historically it had to await two intermediate developments. One of these developments was the increasing emphasis on verification or confirmation, which came with the explicitly so-called verification theory of meaning. The objects of verification or confirmation being statements, this emphasis gave the statement an ascendancy over the word or term as unit of significant discourse. The other development, consequent upon the first, was Russell’s discovery of the concept of incomplete symbols defined in use.
(1961)
This emendation would unquestionably have been welcome to Locke and Hume and Tooke, but historically it had to await an important reorientation in semantics – the reorientation whereby the primary vehicle of meaning came to be seen no longer in the term but in the statement. This reorientation, explicit in Frege (Gottlieb Frege, Foundations of Arithmetic (New York: Philosophical Library, 1950). Reprinted in Grundlagen der Arithmetik (Breslau, 1884) with English translations in parallel. Section 60), underlies Russell’a concept of incomplete symbols defined in use;16b also it is implicit in the verification theory of meaning, since the objects of verification are statements.
Radical reductionism, conceived now with statements as units, sets itself the task of specifying a sense-datum language and showing how to translate the rest of significant discourse, statement by statement, into it. Carnap embarked on this project in the Aufbau.9a</p>
<p>The language which Carnap adopted as his starting point was not a sense-datum language in the narrowest conceivable sense, for it included also the notations of logic, up through higher set theory. In effect it included the whole language of pure mathematics. The ontology implicit in it (i.e., the range of values of its variables) embraced not only sensory events but classes, classes of classes, and so on. Empiricists there are who would boggle at such prodigality. Carnap’s starting point is very parsimonious, however, in its extralogical or sensory part. In a series of constructions in which he exploits the resources of modern logic with much ingenuity, Carnap succeeds in defining a wide array of important additional sensory concepts which, but for his constructions, one would not have dreamed were definable on so slender a basis. Carnap was the first empiricist who, not content with asserting the reducibility of science to terms of immediate experience, took serious steps toward carrying out the reduction.</p>
<p>Even supposing Carnap’s starting point satisfactory, his constructions were, as he himself stressed, only a fragment of the full program. The construction of even the simplest statements about the physical world was left in a sketchy state. Carnap’s suggestions on this subject were, despite their sketchiness, very suggestive. He explained spatio-temporal point-instants as quadruples of real numbers and envisaged assignment of sense qualities to point-instants according to certain canons. Roughly summarized, the plan was that qualities should be assigned to point-instants in such a way as to achieve the laziest world compatible with our experience. The principle of least action was to be our guide in constructing a world from experience.</p>
<p>Carnap did not seem to recognize, however, that his treatment of physical objects fell short of reduction not merely through sketchiness, but in principle. Statements of the form ‘Quality q is at point-instant x; y; z; t’ were, according to his canons, to be apportioned truth values in such a way as to maximize and minimize certain over-all features, and with growth of experience the truth values were to be progressively revised in the same spirit. I think this is a good schematization (deliberately oversimplified, to be sure) of what science really does; but it provides no indication, not even the sketchiest, of how a statement of the form ‘Quality q is at x; y; z; t’ could ever be translated into Carnap’s initial language of sense data and logic. The connective ‘is at’ remains an added undefined connective; the canons counsel us in its use but not in its elimination.</p>
<p>Carnap seems to have appreciated this point afterward; for in his later writings he abandoned all notion of the translatability of statements about the physical world into statements about immediate experience. Reductionism in its radical form has long since ceased to figure in Carnap’s philosophy.</p>
<p>But the dogma of reductionism has, in a subtler and more tenuous form, continued to influence the thought of empiricists. The notion lingers that to each statement, or each synthetic statement, there is associated a unique range of possible sensory events such that the occurrence of any of them would add to the likelihood of truth of the statement, and that there is associated also another unique range of possible sensory events whose occurrence would detract from that likelihood. This notion is of course implicit in the verification theory of meaning.</p>
<p>The dogma of reductionism survives in the supposition that each statement, taken in isolation from its fellows, can admit of confirmation or infirmation at all. My countersuggestion, issuing essentially from Carnap’s doctrine of the physical world in the Aufbau, is that our statements about the external world face the tribunal of sense experience not individually but only as a corporate body. 17b</p>
<p>The dogma of reductionism, even in its attenuated form, is intimately connected with the other dogma: that there is a cleavage between the analytic and the synthetic. We have found ourselves led, indeed, from the latter problem to the former through the verification theory of meaning. More directly, the one dogma clearly supports the other in this way: as long as it is taken to be significant in general to speak of the confirmation and infirmation of a statement, it seems significant to speak also of a limiting kind of statement which is vacuously confirmed, ipso facto, come what may; and such a statement is analytic.</p>
<p>The two dogmas are, indeed, at root identical. We lately reflected that in general the truth of statements does obviously depend both upon extra-linguistic fact; and we noted that this obvious circumstance carries in its train, not logically but all too naturally, a feeling that the truth of a statement is somehow analyzable into a linguistic component and a factual component. The factual component must, if we are empiricists, boil down to a range of confirmatory experiences. In the extreme case where the linguistic component is all that matters, a true statement is analytic. But I hope we are now impressed with how stubbornly the distinction between analytic and synthetic has resisted any straightforward drawing. I am impressed also, apart from prefabricated examples of black and white balls in an urn, with how baffling the problem has always been of arriving at any explicit theory of the empirical confirmation of a synthetic statement. My present suggestion is that it is nonsense, and the root of much nonsense, to speak of a linguistic component and a factual component in the truth of any individual statement. Taken collectively, science has its double dependence upon language and experience; but this duality is not significantly traceable into the statements of science taken one by one.
(1951)
Russell’s concept of definition in use was, as remarked, an advance over the impossible term-by-term empiricism of Locke and Hume. The statement, rather than the term, came with Russell to be recognized as the unit accountable to an empiricist critique.
(1961)
The idea of defining a symbol in use was, as remarked, an advance over the impossible term-by-term empiricism of Locke and Hume. The statement, rather than the term, came with Frege to be recognized as the unit accountable to an empiricist critique.
But what I am now urging is that even in taking the statement as unit we have drawn our grid too finely. The unit of empirical significance is the whole of science.</p>
<p>VI. EMPIRICISM WITHOUT THE DOGMAS
The totality of our so-called knowledge or beliefs, from the most casual matters of geography and history to the profoundest laws of atomic physics or even of pure mathematics and logic, is a man-made fabric which impinges on experience only along the edges. Or, to change the figure, total science is like a field of force whose boundary conditions are experience. A conflict with experience at the periphery occasions readjustments in the interior of the field. Truth values have to be redistributed over some of our statements. Re-evaluation of some statements entails re-evaluation of others, because of their logical interconnections – the logical laws being in turn simply certain further statements of the system, certain further elements of the field. Having re-evaluated one statement we must re-evaluate some others, whether they be statements logically connected with the first or whether they be the statements of logical connections themselves. But the total field is so undetermined by its boundary conditions, experience, that there is much latitude of choice as to what statements to re-evaluate in the light of any single contrary experience. No particular experiences are linked with any particular statements in the interior of the field, except indirectly through considerations of equilibrium affecting the field as a whole.</p>
<p>If this view is right, it is misleading to speak of the empirical content of an individual statement – especially if it be a statement at all remote from the experiential periphery of the field. Furthermore it becomes folly to seek a boundary between synthetic statements, which hold contingently on experience, and analytic statements which hold come what may. Any statement can be held true come what may, if we make drastic enough adjustments elsewhere in the system. Even a statement very close to the periphery can be held true in the face of recalcitrant experience by pleading hallucination or by amending certain statements of the kind called logical laws. Conversely, by the same token, no statement is immune to revision. Revision even of the logical law of the excluded middle has been proposed as a means of simplifying quantum mechanics; and what difference is there in principle between such a shift and the shift whereby Kepler superseded Ptolemy, or Einstein Newton, or Darwin Aristotle?</p>
<p>For vividness I have been speaking in terms of varying distances from a sensory periphery. Let me try now to clarify this notion without metaphor. Certain statements, though about physical objects and not sense experience, seem peculiarly germane to sense experience – and in a selective way: some statements to some experiences, others to others. Such statements, especially germane to particular experiences, I picture as near the periphery. But in this relation of “germaneness” I envisage nothing more than a loose association reflecting the relative likelihood, in practice, of our choosing one statement rather than another for revision in the event of recalcitrant experience. For example, we can imagine recalcitrant experiences to which we would surely be inclined to accommodate our system by re-evaluating just the statement that there are brick houses on Elm Street, together with related statements on the same topic. We can imagine other recalcitrant experiences to which we would be inclined to accommodate our system by re-evaluating just the statement that there are no centaurs, along with kindred statements. A recalcitrant experience can, I have already urged, bc accommodated by any of various alternative re-evaluations in various alternative quarters of the total system; but, in the cases which we are now imagining, our natural tendency to disturb the total system as little as possible would lead us to focus our revisions upon these specific statements concerning brick houses or centaurs. These statements are felt, therefore, to have a sharper empirical reference than highly theoretical statements of physics or logic or ontology. The latter statements may be thought of as relatively centrally located within the total network, meaning merely that little preferential connection with any particular sense data obtrudes itself.</p>
<p>As an empiricist I continue to think of the conceptual scheme of science as a tool, ultimately, for predicting future experience in the light of past experience. Physical objects are conceptually imported into the situation as convenient intermediaries – not by definition in terms of experience, but simply as irreducible posits18b comparable, epistemologically, to the gods of Homer. Let me interject that for my part I do, qua lay physicist, believe in physical objects and not in Homer’s gods; and I consider it a scientific error to believe otherwise. But in point of epistemological footing the physical objects and the gods differ only in degree and not in kind. Both sorts of entities enter our conception only as cultural posits. The myth of physical objects is epistemologically superior to most in that it has proved more efficacious than other myths as a device for working a manageable structure into the flux of experience.
(1951)
Imagine, for the sake of analogy, that we are given the rational numbers. We develop an algebraic theory for reasoning about them, but we find it inconveniently complex, because certain functions such as square root lack values for some arguments. Then it is discovered that the rules of our algebra can be much simplified by conceptually augmenting our ontology with some mythical entities, to be called irrational numbers. All we continue to be really interested in, first and last, are rational numbers; but we find that we can commonly get from one law about rational numbers to another much more quickly and simply by pretending that the irrational numbers are there too.
I think this a fair account of the introduction of irrational numbers and other extensions of the number system. The fact that the mythical status of irrational numbers eventually gave way to the Dedekind- Russell version of them as certain infinite classes of ratios is irrelevant to my analogy. That version is impossible anyway as long as reality is limited to the rational numbers and not extended to classes of them.</p>
<p>Now I suggest that experience is analogous to the rational numbers and that the physical objects, in analogy to the irrational numbers, are posits which serve merely to simplify our treatment of experience. The physical objects are no more reducible to experience than the irrational numbers to rational numbers, but their incorporation into the theory enables us to get more easily from one statement about experience to another.</p>
<p>The salient differences between the positing of physical objects and the positing of irrational numbers are, I think, just two. First, the factor of simplication is more overwhelming in the case of physical objects than in the numerical case. Second, the positing of physical objects is far more archaic, being indeed coeval, I expect, with language itself. For language is social and so depends for its development upon intersubjective reference.</p>
<p>Positing does not stop with macroscopic physical objects. Objects at the atomic level and beyond are posited to make the laws of macroscopic objects, and ultimately the laws of experience, simpler and more manageable; and we need not expect or demand full definition of atomic and subatomic entities in terms of macroscopic ones, any more than definition of macroscopic things in terms of sense data. Science is a continuation of common sense, and it continues the common-sense expedient of swelling ontology to simplify theory.</p>
<p>Physical objects, small and large, are not the only posits. Forces are another example; and indeed we are told nowadays that the boundary between energy and matter is obsolete. Moreover, the abstract entities which are the substance of mathematics – ultimately classes and classes of classes and so on up – are another posit in the same spirit. Epistemologically these are myths on the same footing with physical objects and gods, neither better nor worse except for differences in the degree to which they expedite our dealings with sense experiences.</p>
<p>The over-all algebra of rational and irrational numbers is underdetermined by the algebra of rational numbers, but is smoother and more convenient; and it includes the algebra of rational numbers as a jagged or gerrymandered part.19b Total science, mathematical and natural and human, is similarly but more extremely underdetermined by experience. The edge of the system must be kept squared with experience; the rest, with all its elaborate myths or fictions, has as its objective the simplicity of laws.</p>
<p>Ontological questions, under this view, are on a par with questions of natural science.20b Consider the question whether to countenance classes as entities. This, as I have argued elsewhere,10a21b is the question whether to quantify with respect to variables which take classes as values. Now Carnap [“Empiricism, semantics, and ontology,” Revue internationale de philosophie 4 (1950), 20-40.] has maintained11a that this is a question not of matters of fact but of choosing a convenient language form, a convenient conceptual scheme or framework for science. With this I agree, but only on the proviso that the same be conceded regarding scientific hypotheses generally. Carnap has recognized12a that he is able to preserve a double standard for ontological questions and scientific hypotheses only by assuming an absolute distinction between the analytic and the synthetic; and I need not say again that this is a distinction which I reject. 22b
(1951)
Some issues do, I grant, seem more a question of convenient conceptual scheme and others more a question of brute fact.
The issue over there being classes seems more a question of convenient conceptual scheme; the issue over there being centaurs, or brick houses on Elm Street, seems more a question of fact. But I have been urging that this difference is only one of degree, and that it turns upon our vaguely pragmatic inclination to adjust one strand of the fabric of science rather than another in accommodating some particular recalcitrant experience. Conservatism figures in such choices, and so does the quest for simplicity.</p>
<p>Carnap, Lewis, and others take a pragmatic stand on the question of choosing between language forms, scientific frameworks; but their pragmatism leaves off at the imagined boundary between the analytic and the synthetic. In repudiating such a boundary I espouse a more thorough pragmatism. Each man is given a scientific heritage plus a continuing barrage of sensory stimulation; and the considerations which guide him in warping his scientific heritage to fit his continuing sensory promptings are, where rational, pragmatic.</p>
<p>Notes
1a. Much of this paper is devoted to a critique of analyticity which I have been urging orally and in correspondence for years past. My debt to the other participants in those discussions, notably Carnap, Church, Goodman, Tarski, and White, is large and indeterminate. White’s excellent essay “The Analytic and the Synthetic: An Untenable Dualism,” in John Dewey: Philosopher of Science and Freedom (New York, 1950), says much of what needed to be said on the topic; but in the present paper I touch on some further aspects of the problem. I am grateful to Dr. Donald L. Davidson for valuable criticism of the first draft.</p>
<p>2a. See White, “The Analytic and the Synthetic: An Untenable Dualism,” John Dewey: Philosopher of Science and Freedom (New York: 1950), p. 324.</p>
<p>1b. See “On What There Is,” p. 9.</p>
<p>3a. R. Carnap, Meaning and Necessity (Chicago, 1947), pp. 9 ff.; Logical Foundations of Probability (Chicago, 1950), pp. 70 ff.</p>
<p>2b. See “On What There Is,” p. 10.</p>
<p>4a. This is cognitive synonymy in a primary, broad sense. Carnap (Meaning and Necessity, pp. 56 ff.) and Lewis (Analysis of Knowledge and Valuation [La Salle, Ill., 1946], pp. 83 ff.) have suggested how, once this notion is at hand, a narrower sense of cognitive synonymy which is preferable for some purposes can in turn be derived. But this special ramification of concept-building lies aside from the present purposes and must not be confused with the broad sort of cognitive synonymy here concerned.</p>
<p>3b. See “On What There Is”, p. 11f, and “The Problem of Meaning in Linguistics,” p. 48f.</p>
<p>5a. See, for example my Mathematical Logic (New York, 1949; Cambridge, Mass., 1947), sec. 24, 26, 27; or Methods of Logic (New York, 1950), sec. 37 ff.</p>
<p>4b. Rudolf Carnap, Meaning and Necessity (Chicago: University of Chicago Press, 1947), pp. 9ff; Logical Foundations of Probability (Chicago: University of Chicago Press, 1950).</p>
<p>6a. The ‘if and only if’ itself is intended in the truth functional sense. See Carnap, Meaning and Necessity, p. 14.</p>
<p>5b. According to an important variant sense of ‘definition’, the relation preserved may be the weaker relation of mere agreement in reference; see “Notes on the Theory of Reference,” p. 132. But, definition in this sense is better ignored in the present connection, being irrelevant to the question of synonymy.</p>
<p>7a. The doctrine can indeed be formulated with terms rather than statements as the units. Thus C. I. Lewis describes the meaning of a term as “a criterion in mind, by reference to which one is able to apply or refuse to apply the expression in question in the case of presented, or imagined things or situations” (Carnap, Meaning and Necessity, p. 133.).</p>
<p>6b. Cf. C.I. Lewis, A Survey of Symbolic Logic (Berkeley, 1918), p. 373.</p>
<p>8a. John Horne Tooke, The Diversions of Purly (London, 1776; Boston, 1806), I, ch. ii.</p>
<p>7b. This is cognitive synonymy in a primary, broad sense. Carnap (Meaning and Necessity, pp. 56 ff.) and Lewis (Analysis of Knowledge and Valuation [La Salle, Ill., 1946], pp. 83 ff.) have suggested how, once this notion is at hand, a narrower sense of cognitive synonymy which is preferable for some purposes can in turn be derived. But this special ramification of concept-building lies aside from the present purposes and must not be confused with the broad sort of cognitive synonymy here concerned.</p>
<p>9a. R. Carnap, Der logische Aufbau der Welt (Berlin, 1928).</p>
<p>8b. Pp. 81ff, “New Foundations for Mathematical Logic,” contains a description of just such a language, except that there happens to be just one predicate, the two-place predicate ‘e’.</p>
<p>10a. For example, in “Notes on Existence and Necessity,” Journal of Philosophy, 11 (1943), 113-127.</p>
<p>9b. See “On What There Is,” pp. 5-8; see also “New Foundations for Mathematical Logic,” p. 85f; “Meaning and Existential Inference,” p. 166f.</p>
<p>11a. Carnap, “Empiricism, Semantics, and Ontology,” Revue internationale de philosophie, 4 (1950), 20-40.</p>
<p>10b. See “New Foundations for Mathematical Logic,” p. 87.</p>
<p>12a. Carnap, “Empiricism, Semantics, and Ontology,” p. 32.</p>
<p>11b. On such devices see also “Reference and Modality.”</p>
<p>12b. This is the substance of Quine, Mathematical Logic (1940; rev. ed., 1951).</p>
<p>13b. The ‘if and only if’ itself is intended in the truth functional sense. See R. Carnap, Meaning and Necessity (1947), p. 14.</p>
<p>14b. The foregoing paragraph was not part of the present essay as originally published. It was prompted by Martin, (R. M. Martin, “On ‘analytic’,” Philosophical Studies 3 (1952): 42-47.</p>
<p>15b. The doctrine can indeed be formulated with terms rather than statements as the units. Thus Lewis describes the meaning ot a term as “a criterion in mind bv reference to which one is able to apply or refuse to apply the expression in question in the case of presented, or imagined, things or situations” ([2], p. 133). – For an instructive account of the vicissitudes of the verification theory of meaning, centered however on the question of meaninfulness rather than synonymy and analyticity, see Hempel.</p>
<p>16b. See “On What There Is,”, p. 6.</p>
<p>17b. This doctrine was well argued by Pierre Duhem, La Theorie physique: son objet et sa structure (Paris, 1906): 303-328. Or see Armand Lowinger, The Methodology of Pierre Duhem (New York: Columbia University Press, 1941): 132-140.</p>
<p>18b. Cf. pp. 17f “On What There Is.”</p>
<p>19b. Cf. p. 18 “On What There Is.”</p>
<p>20b. “L’ontologie fait corps avec la science elle-mene et ne peut en etre separee.” Meyerson, p. 439.</p>
<p>21b. “On What There Is,” pp. 12f; “Logic and the Reification of Universals,” pp. 102ff.a</p>
<p>22b. For an effective expression of further misgivings over this distinction, see White “The Analytic and the Synthetic: An Untenable Dualism.”</p>Sina HazratpourQuine's Empiricism about mathematicsHeidegger’s Parmenides2021-02-19T00:00:00+00:002021-02-19T00:00:00+00:00https://sinhp.github.io/posts/2021/02/heidegger_parmenides<section>
<h2 id="epigraph"></h2>
<div class="epigraph">
<blockquote>
<p>The water of the river flowing in the field of λήθη eludes all containment and itself effectuates only the one withdrawal letting everything escape and thus concealing everything. [...] "Philosophy" is accordingly not a mere dealing with universal concepts on the part of thinking, to which one can dedicate oneself or not without there occurring anything essential. Philosophy means to be addressed by Being itself. Philosophy is in itself the basic mode in which man comports himself to beings in the midst of beings. Men who lack philosophy are without insight. They deliver themselves over to what happens to appear and likewise to what happens to disappear. They are at the mercy of the withdrawal and the concealment of beings. They drink beyond the measure of the water of the river "Carefree".</p>
<footer>Martin Heidegger, Parmenides, The Third Directive</footer>
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<p>The Winter Semester 1942–43 lecture course was originally announced under the title Parmenides and Heraclitus. It contains a long meditation on ἀλήθεια by way of interpretation of the famous poem of Parmenides.</p>
<p>Heidegger approaches ἀλήθεια from four different directions. Ἀλήθεια means unconcealment. We can read unconcealment as unconcealment and as un-concealment.</p>
<p>When we read un-concealment, ἀλήθεια means the coming into its own of truth and of concealment as sheltering this truth.
When we read un-concealment, it becomes clear that the Greeks discovered in the essential swaying (Wesung) and dynamic of truth the negation of concealment.
The third indication is the relation between ἀλήθεια and lèthè, forgottenness. Since truth has to be wrestled from unconcealment, it always risks falling back into forgottenness. Parmenides names the unconcealment of being, and yet this simple truth would soon be forgotten.
The fourth indication names the relation between unconcealment and clearing. Ἀλήθεια clears the open within this free region; entities can come to presence in the way they look, εἶδος. When we free ourselves form the presence of entities, we may spring into the abground and recollect the truth of being. We can then become aware of the difference between being and entities.
The saying of Parmenides says the beginning of the still-concealed withdrawal of the truth of being. It names the belonging together of human beings and ἀλήθεια.</p>
<p>From Heidegger Gesamtausgabe
Complete works and concomitant content on beyng.com</p>
<p>EINLEITUNG</p>
<p>VORBEREITENDE BESINNUNG AUF DEN NAMEN UND DAS WORT ΑΛΗΘΕΙΑ UND SEIN GEGENWESEN. ZWEI WEISUNGEN DES UBERSETZENDEN WORTES</p>
<p>ΑΛΗΘΕΙΑ</p>
<p>§ 1. Die Göttin » Wahrheit«. Parmenides, I, 22—32</p>
<p>a) Das gewöhnliche Sichauskennen und das wesentliche</p>
<p>Wissen. Die Absage an das Geläufige des »Lehrgedichtes« durch das Aufmerken auf den Anspruch des Anfangs</p>
<p>Παρμενίδης και Ηράκλειτος, Parmenides und Heraklit, Zeitgenossen in den Jahrzehnten zwischen 540/460, heißen die beiden Denker, die in einer einzigen Zusammengehörigkeit am Beginn des abendländischen Denkens das Wahre denken. Das Wahre denken heißt: das Wahre in seinem Wesen erfahren und in solcher Wesenserfahrung die Wahrheit des Wahren wissen. Nach der Zeitrechnung sind seit dem Beginn des abendländischen Denkens zweitausend und fünfhundert Jahre vergangen. Was im Denken der beiden Denker gedacht ist, wird vom Vergehen der Jahre und Jahrhunderte niemals berührt. Diese Unberührbarkeit durch die zehrende Zeit gilt aber keineswegs deshalb, weil das Gedachte, das diese Denker denken mußten, seitdem irgendwo an sich an einem überzeitlichen Ort aufbewahrt wird als das sogenannte »Ewige«. Vielmehr ist das Gedachte dieses Denkens gerade das eigentlich Geschichtliche, was aller nachfolgenden Geschichte vorauf- und d. h. vorausgeht. Das also Voraufgehende und alle Geschichte Bestimmende nennen wir das Anfängliche. Weil es nicht in einer Vergangenheit zurück-, sondern dem Kommenden vorausliegt, macht sich das Anfängliche immer einmal wieder einem Zeitalter eigens zum Geschenk.</p>
<p>Der Anfang ist das, was in der wesenhaften Geschichte zuletzt kommt. Für ein Denken allerdings, das nur die Form des Rechnens kennt, bleibt der Satz: »Der Anfang ist das Letzte«, ein Widersinn. Zuerst freilich, an seinem Beginn, erscheint der Anfang in einer eigentümlichen Verhüllung. Daher stammt das Merkwürdige, daß das Anfängliche leicht für das Unvollkommene, Unfertige, Grobe gilt. Man nennt es auch »das Primitive«. So entsteht dann die Meinung, die Denker vor Platon und Aristoteles seien noch »primitive Denker«. Allerdings ist nicht jeder Denker im Beginn des abendländischen Denkens auch schon ein anfänglicher Denker. Der erste anfängliche Denker heißt Anaximander.</p>
<p>Die beiden anderen und mit ihm die einzigen sind Parmenides und Heraklit. Daß wir diese drei Denker als die erstanfänglichen vor allen anderen Denkern des Abendlandes auszeichnen, erweckt den Eindruck der Willkür. Wir besitzen in der Tat auch keine Beweismittel, die genügten, die genannte Auszeichnung unmittelbar zu begründen. Dazu ist nötig, daß wir erst zu diesen anfänglichen Denkern in einen echten Bezug gelangen. Das soll in den Stunden dieser Vorlesung versucht werden.</p>
<p>Im Ablauf der Zeitalter der abendländischen Geschichte entfernt sich das nachkommende Denken nicht nur in der Zeitfolge von seinem Beginn, sondern auch und vor allem in dem, was gedacht wird, von seinem Anfang. Die folgenden Menschengeschlechter werden dem frühen Denken mehr und mehr entfremdet. Zuletzt ist der Abstand so groß, daß der Zweifel sich regt, ob ein späteres Zeitalter überhaupt noch die frühesten Gedanken wieder zu denken vermöge. Zu diesem Zweifel gesellt sich der andere, ob dies Vorhaben, gesetzt es sei möglich, noch irgend einen Nutzen bringe. Wozu sollen wir auf den fast ausgelöschten Spuren eines längst vergangenen Denkens umherirren? Die Zweifel an der Möglichkeit und am</p>Sina HazratpourPFMRG: The First Phase2021-02-17T00:00:00+00:002021-02-17T00:00:00+00:00https://sinhp.github.io/posts/2021/02/pfmrg_i<section>
<h2 id="epigraph"></h2>
<div class="epigraph">
<blockquote>
<p>Wir tragen empfängliche Herzen im Busen,</p>
<p>Wir geben uns willig der Täuschung hin!</p>
<p>Drum weilet gern, ihr holden Musen,</p>
<p>Bei einem Volke mit offenem Sinn.</p>
<footer>Ludwig van Beethoven, “Die Ruinen von Athen”</footer>
</blockquote>
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<section>
</section>
<h2 id="pfmrg-the-reading-resources-for-the-first-phase">PFMRG: The Reading Resources For The First Phase</h2>
<p>In the first few sessions we are going to learn about basic questions of philosophy of mathematics and get ourselves familiar with formal logic and formal proofs. Our discussions follow two tracks.</p>
<section>
</section>
<h3 id="track-i-basic-questions-of-philosophy-of-mathematics">Track I: Basic Questions Of Philosophy Of Mathematics</h3>
<p>I have picked two introductory resources:</p>
<ul>
<li>Øystein Linnebo. Philosophy of Mathematics, Princeton University Press, 2017</li>
<li>David Corfield. Towards a Philosophy of Real Mathematics Cambridge University Press, 2006</li>
</ul>
<p>Linnebo’s book will be our main source. It is an introductory textbook to the philosophy of mathematics. The advanatge of this book to many other books at the same introductory level is its achievement of explaining philosophical issues in a clear and simple language. Also, it does not require a great deal of background in logic or mathematics, but it does not evade <span>mathematical technicalities either <label for="sn-merging" class="margin-toggle sidenote-number"></label></span><input type="checkbox" id="sn-merging" class="margin-toggle" /><span class="sidenote"><em>It is not easy to achieve this! I think Linnebo does a good job of merging logic and mathematics with their underpinning philosophies: just enough to serve the purposes of the book; after all, this is not a mathematics book.</em></span>: Since our reading group comprises of people with hugely varying degree of mathematical maturity, this makes the book a great fit for us. Finally, the book covers all important movements and trends in philosophy of mathematics with the exception of Pragmatism (Peirce, Ramsey, and others).</p>
<p>Second, the book does not hide the great prominence it gives to Frege and his philosophy of mathematics throughout the chapters. I personally appreciate this honesty and I wish I could see it more in other publications.</p>
<p>We will read and discuss almost all of its chapters. To be prepared for our first few sessions I recommend reading chapters 1-4. In below, I have included additional supplementary materials for these chapters.</p>
<p>Corfield’s book is complementary to Linnebo’s book; It gives a pluralistic point of view to the philosophy and foundations of mathematics. It is only supplementary for the first phase. I think Corfield’s book will be a great source of ideas when we move on to the second phase of our reading group.</p>
<section>
</section>
<h3 id="supplementary-reading-materials-for-track-i">Supplementary Reading Materials For Track I</h3>
<ul>
<li>
<p>We shall talk about Frege’s “context principle”. Linnebo explains it in the book very well but the discussion of it is rather short. However, there is a very nice talk by him about the context principle: <a href="https://www.youtube.com/watch?v=l_g44OP-6UU&t=905s&ab_channel=Facolt%C3%A0diFilosofiaUniSR">The role of Frege’s Context Principle in his epistemology of mathematics, Øystein Linnebo</a></p>
</li>
<li>
<p>A great complementary source is Awodey and Reck’s <a href="https://academic.oup.com/philmat/article-abstract/13/2/225/1536961">Frege’s Lectures on Logic: Carnap’s Student Notes, 1910–1914</a></p>
</li>
<li>
<p>Steve Awodey, A. W. Carus. <a href="https://www.cambridge.org/core/books/cambridge-companion-to-logical-empiricism/turning-point-and-the-revolution-philosophy-of-mathematics-in-logical-empiricism-from-tractatus-to-logical-syntax/083CAACE00CFC136E311F062E5AF7291">The Turning Point and the Revolution: Philosophy of Mathematics in Logical Empiricism from Tractatus to Logical Syntax</a>. Pages 6 & 7 are extremely relevanet to what we discuss regarding “mathematical objects and mathematical objectivity” in Frege’s philosophy.</p>
</li>
</ul>
<section>
</section>
<h3 id="track-ii-introduction-to-formal-logic--formal-proofs">Track II: Introduction To Formal Logic & Formal Proofs</h3>
<p>For this track we have the following reading materials:</p>
<ul>
<li><a href="https://www.princeton.edu/~fraassen/Formal%20Semantics%20and%20Logic.pdf">Bas van Fraassen. Formal semantics and logic</a>. Macmillan Co. New York 1971, newly digitized version 2016.</li>
</ul>
<p>van Fraassen’s book is based on his lectures in logic at Yale 1966–1968 and Indiana 1969–1970. These lectures were intended specifically for philosophy students. We will have a selected reading of chapters 1-3.</p>
<ul>
<li>Aristotle. Prior Analytics, circa 350 BCE
Translated by <a href="http://classics.mit.edu/Aristotle/prior.htm">A. J. Jenkinson</a></li>
<li>Other translations:
<ul>
<li>Aristotle, Prior Analytics, translated by Robin Smith, Indianapolis: Hackett, 1989.</li>
<li>Aristotle, Prior Analytics Book I, translated by Gisela Striker, Oxford: Clarendon Press 2009</li>
</ul>
</li>
</ul>
<p>Aristotle’s Prior Analytics is the first time in history when Logic is scientifically investigated. The book is famously about deductive reasoning, known as syllogism. This will be our first introduction to deductive reasoning.</p>
<h3 id="supplementary-reading-materials-for-track-ii">Supplementary Reading Materials For Track II</h3>
<ul>
<li>
<p>Brentano’s doctoral dissertation
Brentano saw himself not merely as a historical scholar of
Aristotle, but as his intimate disciple. Brentano finds that (in Aristotle’s view) being in the sense
of the categories, in particular substantial being, is the most basic; all other modes, potential and actual being, being in the sense of the true, etc., stand to it in a relation of well-foundedanalogy.
Many of his mature views are prepared in this work. For example his discussion of being in the sense of being true appears to be the foundation of his later non-propositional
theory of judgment.</p>
</li>
<li>C. S. Peirce. Collected Papers of Charles Sanders Peirce, volume 7. Harvard University Press, Cambridge,
MA, 1935. Edited by C. Hartshorne, P. Weiss, and A.W. Bucks. Paragraph 218.</li>
<li><a href="http://www.jfsowa.com/pubs/egtut.pdf">Peirce Existential Graphs</a></li>
<li>Jeremy Avigad. Reliability of mathematical inference. Synthese, pages 1–23, 2020</li>
<li>I. Lakatos. Proofs and Refutations. The Logic of Mathematical Discovery. Cambridge University Press. 2015</li>
<li>William Thurston. <a href="https://arxiv.org/pdf/math/9404236.pdf">On Proof and Progress in Mathematics</a>. 1994.</li>
</ul>
<section>
</section>
<h2 id="formal-logic-begins-with-aristotle">Formal Logic Begins With Aristotle</h2>
<p><span class="newthought">The first paragraph of Prior Analytics <label for="sn-striker-tr" class="margin-toggle sidenote-number"></label></span><input type="checkbox" name="sn-striker-tr" class="margin-toggle" /><span class="sidenote"><em>From the beautiful translation by Gisela Striker</em></span> gives a pretty good summary of what is in rest of the book:</p>
<blockquote>
<p><em>First, to say about what and of what this is an investigation: it is about demonstration and of demonstrative science. Then, to define what is
a premiss, what is a term, and what a syllogism, and which kind of
syllogism is perfect and which imperfect. After that, what it is for
this to be or not to be in that as in a whole, and what we mean by ‘to be predicated of all’ or ‘of none’.</em></p>
</blockquote>
<p>Aristotle then continues to define <q>syllogism</q>.</p>
<blockquote>
<p><em>A syllogism is an argument in which, certain things being posited, something other than what was laid down results by necessity because these things are so. By ‘because these things are so’ I mean that it results through these, and by ‘resulting through these’ I mean that no term is required from outside for the necessity to come about.</em></p>
</blockquote>
<p><a href="https://philpapers.org/archive/CORAND.pdf">Corcoran-Aristotle’S Natural Deduction System</a></p>
<p>It will be seen that Aristotie’s theory of deduction contains a self-sufficient natural deduction system which presupposes no other logic. Perhaps the reason that Aristotle’s theory of deduction has been overlooked is that it differs radically from many of the ‘standard’ modern systems. It has no axioms, it involves no truth-functional combinations and it lacks both the explicit and implicit quantifiers (in the modern sense). … My opinion is this: if the Lukasiewicz view is correct then
Aristotle cannot be regarded as the founder of logic. Aristotle would merit this title no more than Euclid, Peano, or Zermelo insofar as these men are regarded as founders, respectively, of axiomatic geometry, axiomatie arithmetic and axiomatic set theory. (AristotIe would be merely the founder of ‘the axiomatic theory of universals’.) Each ofthe former three
men set down an axiomatization of a body of information without explicitly developing the underlying logic. That is, each of these men put
down axioms and regarded as theorems of the system the sentences obtainable from the axioms by logical deductions but without bothering to
say what a logical deduction is.</p>
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<p>You ask me, which of the philosophers' traits are idiosyncracies? For example: their lack of historical sense, their hatred of becoming, their Egypticism. They think that they show their respect for a subject when they dehistoricize it — when they turn it into a mummy. </p>
<footer>Friedrich Nietzsche, “The Twilight of the Idols”</footer>
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>_You ask me, which of the philosophers' traits are idiosyncracies? For example: their lack of historical sense, their hatred of becoming, their Egypticism. They think that they show their respect for a subject when they dehistoricize it — when they turn it into a mummy.
Friedrich Nietzsche, The Twilight of the Idols_
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<p>This article is about a <span>Philosophy and Foundations of Mathematics Reading Group (PFMRG)<label for="sn-pfmrg" class="margin-toggle sidenote-number"></label></span><input type="checkbox" id="sn-pfmrg" class="margin-toggle" /><span class="sidenote"><em>I am leading this group. It is hosted on Falafe server on
<a href="https://discord.com/" target="_blank">Discord</a> where we have well over a hundred members. The running language is Persian. We meet bi-weekly, and each session will last about two hours (30-45 minutes for presentation, the rest for the Q&A). Most members (mostly PhDs and postdocs in sciences, arts, philosophy, and engineering) have keen interests in philosophical matters, but have little or no previous experience in logic or mathematics. In a way, this is even a better occasion to think the original and the historical, the genuinely histoical, preceding and thereby anticipating all successive history of logic and philosophy of mathematics. The recording of the talks are found in this <a href="https://youtube.com/playlist?list=PLLSwxwJoqOFFaNVfQoxxApcM15Et4rgiP">channel</a>. </em>
<br />
</span>
which is inaugurated in mid February 2021. The goals of this group and the themes of the presentations are discussed in below. The main characteristics of the reading group are the diversity of topics we shall discuss as well as covering both historical and modern aspects.
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</p>
<section>
</section>
<h2 id="goals--hopes">Goals & Hopes</h2>
<p>The main goal of this reading group is to understand how close the human has come to the point where his machines can do mathematics for him. Some mathematicians, such the Fields medalists Paul Cohen, Tim Gowers, Maxim Kontsevich, and Vladimir Voedodsky have all at some point predicted that computers will be able to out-reason mathematicians in the 21st century. But is that really plausible?</p>
<p>To make any sound judgment on this question, we need to better understand</p>
<ol>
<li>
<p>What is (and what should be regarded as) mathematical progress?</p>
</li>
<li>
<p>What is (and what should be regarded as) mathematical proof? Do we always need formal proofs to do mathematics? (c.f. theorems vs <a href="https://en.wikipedia.org/wiki/Porism">porisms</a> )</p>
</li>
<li>
<p>What is the role of human intuition in mathematical reasoning and in mathematical proofs? Where do new ideas in mathematics come from?</p>
</li>
<li>
<p>What is the role of natural language in mathematics and how different is it from the mathematical languages in which the proofs are written? What are the limits of our mathematical languages compared to our natural language where we reason about mathematics at a meta level? And could these limits be problematic for the computers to do mathematics equally or even better than humans?</p>
</li>
<li>
<p>What is the machine mathematics? What are automated theorem provers (ATPs) and interactive theorem provers (ITPS) and what are their differences? Can the advances in AI, ML, and neuroscience bridge the gap between ATP+ITP and the human mind when it comes to doing (creating/discovering) interesting new mathematics?</p>
</li>
<li>
<p>Will we have two mathematics in future: the machine mathematics, and the human mathematics? And how do we, humans, react to such a situation? Will we simply accept the proofs generated by computers which conceivably could not be understood by any human being at all? Will such proofs have the same status as human generated proofs</p>
</li>
</ol>
<p>These are pressing questions and we need to ransack various philosophies and foundations of mathematics that have been thought before us; they give us radically different insights to the questions above and presents us with cast territories of philosophical though. In the first session of our reading group, I will give an introductory talk charting out a basic map of these territories, and establishing some reading resources that get us going.</p>
<p>To put some of the discussions above in context, <a href="https://mathoverflow.net/questions/43690/whats-a-mathematician-to-do">here</a> is the point of view of the geometrician Bill Thurston
regarding the activity of mathematicians:</p>
<blockquote>
<p><em>The product of mathematics is clarity and understanding. Not theorems, by themselves. Is there, for example any real reason that even such famous results as Fermat’s Last Theorem, or the Poincaré conjecture, really matter? Their real importance is not in their specific statements, but their role in challenging our understanding, presenting challenges that led to mathematical developments that increased our understanding.</em></p>
</blockquote>
<blockquote>
<p><em>I think of mathematics as having a large component of psychology, because of its strong dependence on human minds. Dehumanized mathematics would be more like computer code, which is very different. Mathematical ideas, even simple ideas, are often hard to transplant from mind to mind. Mathematical ideas, even simple ideas, are often hard to transplant from mind to mind. There are many ideas in mathematics that may be hard to get, but are easy once you get them. Because of this, mathematical understanding does not expand in a monotone direction.</em></p>
</blockquote>
<p>And he continues</p>
<blockquote>
<p><em>Our understanding frequently deteriorates as well. There are several obvious mechanisms of decay. The experts in a subject retire and die, or simply move on to other subjects and forget. Mathematics is commonly explained and recorded in symbolic and concrete forms that are easy to communicate, rather than in conceptual forms that are easy to understand once communicated. Translation in the direction <code>conceptual -> concrete and symbolic</code> is much easier than translation in the reverse direction, and symbolic forms often replaces the conceptual forms of understanding. And mathematical conventions and taken-for-granted knowledge change, so older texts may become hard to understand.</em></p>
</blockquote>
<section>
</section>
<h2 id="which-philosophies-which-foundations">Which Philosophies? Which Foundations?</h2>
<p>Here is a common sense and self-evident proposal:</p>
<blockquote>
<p><em>What mathematicians think and do should be important for the philosophy of mathematics.</em></p>
</blockquote>
<p>Now, if you are not familiar with the current state of affairs in the philosophy of mathematics in academia, you are for a hard surprise that the proposal above could not be further from the reality of the most of academic activities in philosophy of mathematics. Not in this reading group. We do not want to confine ourselves with the narrow interests of the academic philosophy of mathematics. Here we tend to think of mathematics as a rich network of ideas and their formal embodiements, therefore we will be concerned with the history and development of ideas, concepts, and structures in various branches of mathematics rather than reducing the content of our philosophical interest either to set theory or arithmetic. Therefore, a basic knowledge of mathematics is necessary as a background for the philosophy of mathematics. To get started, perhaps the best introduction is Saunders MacLane’s <a href="https://www.springer.com/gp/book/9781461293408">Mathematics Form and Function</a>.</p>
<p>In particular, we shall try to ask questions such as</p>
<blockquote>
<p><em>When do we call a kind of knowledge mathematical? How does mathematical knowledge grow? What is mathematical progress? What makes some mathematical ideas (or theories) better than others? What is mathematical explanation? etc</em></p>
</blockquote>
<p>I believe in epistemological anarchism, that is, if needed, we have to be receptive to ideas from the most disparate and apparently far-flung domains, even at the price of losing the comfort of our nests. Therefore, in this reading group we will remain open to all ideas and philosophies as long as they are helpful to our goals and hopes discussed in above. May this group be a place for deep learning (and deep unlearning/ unindoctorination) as well as unashamedly free speculations. We will review the main tenets of the following philosophies and foundations of mathematics:</p>
<ul>
<li>Constructivism and Intuitionism</li>
<li>Structuralism</li>
<li>Formalism</li>
<li>Fictionalism</li>
<li>Cognitivism</li>
</ul>
<section>
</section>
<h2 id="upcoming-talks">Upcoming Talks</h2>
<p>(2) Introduction to formal logic and formal proofs: From Aristotle’s Prior Analytics to Frege’s Begriffsschrift]: <a href="https://sinhp.github.io/files/Phil/philos_math/pfmrg-slides/index.html#/apriori_aposteriori">slides</a></p>
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<h2 id="past-talks">Past Talks</h2>
<p>(1) Introduction to Logicism an Frege’s Begriffsschrift, <a href="https://sinhp.github.io/files/Phil/philos_math/pfmrg-slides/index.html#/apriori_aposteriori">slides</a>, <a href="https://youtu.be/YEO1n6aLi7E">video recording</a></p>
<p>(0) The initial meeting: Introduction to Kant’s philosophy of mathematics, <a href="/files/Phil/philos_math/pfmrg-slides/index.html">slides</a>, <a href="https://youtu.be/HGZ8-58HS3w">video recording</a></p>
<p>(-1) <a href="/files/Phil/philos_math/intuitionism_roots_part_ii.pdf">The Roots of Constructivism and Intuitionism in Mathematics - Part II</a>
This talk introduced various aspects of intuitionism and its vast differences from the classicial foundation of mathematics (ZFC set theory built on top of the classicial logic). Topics discussed included Brouwer’s writings, BHK interpretation, the principle of excluded middle, the axiom of choice, the continuum, the role of real numbers in classicial and intuionistic mathematics, and the creating mind’s perception of time.</p>
<p>(-2) <a href="/files/Phil/philos_math/intuitionism_roots_part_i.pdf">The Roots of Constructivism and Intuitionism in Mathematics - Part I</a>
This talk mostly centered around the role of foundations, and an informal discussion of various foundations of mathematics</p>
<section>
<h2 id="epigraph"></h2>
<div class="epigraph">
<blockquote>
<p>Historically what has underlied the growth of our cognitive capacity for generating and understanding new mathematical concepts is basically the recognition of our increasing tolerance for thinking the insane.</p>
<footer></footer>
</blockquote>
</div>
</section>
<h2 id="resources-and-learning-materials">Resources and Learning Materials</h2>
<p class="sans">In below, I list some of the references which I personally found interesting, intriguing, and edifying. Occasionally, we will discuss some of them, but we do not limit ourselves to them.</p>
<h3 id="i-have-no-clue-what-the-philosophy-of-mathematics-is-about-where-should-i-begin">I have no clue what the philosophy of mathematics is about. Where should I begin?</h3>
<ul>
<li>At night before sleep:
<ul>
<li><a href="https://www.logicomix.com/en/index.html">LOGICOMIX</a></li>
<li><a href="https://www.c82.net/euclid/">BYRNE’S EUCLID</a></li>
<li>Noson S. Yanofsky. The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us.</li>
<li><a href="https://arxiv.org/pdf/math/9404236.pdf">On Proof and Progress in Mathematics</a>, See also<br />
<a href="https://mathoverflow.net/questions/43690/whats-a-mathematician-to-do">What’s a mathematician to do?</a></li>
<li>Saunders MacLane. Mathematics Form and Function</li>
<li>Robert L. Long. Remarks on the history and philosophy of mathematics. The American Mathematical Monthly, 93(8):609–619, 1986.</li>
</ul>
</li>
<li>After mid-day coffee:
<ul>
<li>Mark Colyvan. An Introduction to the Philosophy of Mathematics</li>
<li>David Corfield. Towards a Philosophy of Real Mathematics</li>
<li>Øystein Linnebo. Philosophy of Mathematics</li>
<li>Penrose, L. S. and Penrose, R. Impossible Objects, a Special Kind of Illusion, 1958. British Journal of Psychology, 49: 31-33.</li>
<li>Where does mathematics come from?
<ul>
<li><a href="https://www.kavlifoundation.org/science-spotlights/brain-or-universe-%E2%80%93-where-does-math-come">The Brain or the Universe – Where Does Math Come From?, A Talk at Kavli Foundation</a></li>
<li>Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being
The authors claim that the mathematics we used to describe as disembodied is in fact embodied. Humans use their bodies, mind, and brain to both form and understand mathematics. All mathematical content resides in embodied ideas and many of the most basic, as well as the most sophisticated, mathematical ideas are metaphorical.</li>
</ul>
</li>
<li>Hermann Weyl. Mind and nature: Selected writings on philosophy, mathematics and physics. 2002. Edited and with an introduction by Peter Pesic.</li>
<li>Colin McLarty. The rising sea: Grothendieck on simplicity and generality</li>
<li>Howard Stein. Logos, Logic, and Logistike. Phillip Kitcher, eds., History and Philosophy of Modern Mathematics. University of Minnesota Press 238–259, 1988.</li>
</ul>
</li>
<li>While waking up in the morning or chilaxing you can watch:
<ul>
<li>Ray Monk. <a href="https://www.youtube.com/watch?v=bqGXdh6zb2k&ab_channel=PhilosophyOverdose">Intro to the Philosophy of Mathematics</a></li>
<li>Denis Bonnay. <a href="https://www.youtube.com/watch?v=5LovB0kbkJc&ab_channel=SeriousScience">Logic and Mathematics</a></li>
<li>Stephen Read. <a href="https://www.youtube.com/watch?v=uDTUgC48GlY&ab_channel=SeriousScience">Logical Paradoxes</a></li>
<li>Hartry Field. Logic, Normativity, and Rational Revisability, John Locke Lectures in Philosophy.
Part <a href="https://podcasts.ox.ac.uk/2008-lecture-1-puzzle-about-rational-revisability">I</a>, <a href="https://podcasts.ox.ac.uk/2008-lecture-2-what-normative-role-logic">II</a>, <a href="https://podcasts.ox.ac.uk/2008-lecture-3-case-rational-revisability-logic">III</a>, <a href="https://podcasts.ox.ac.uk/2008-lecture-4-really-revising-logic">IV</a>,
<a href="https://podcasts.ox.ac.uk/2008-lecture-5-epistemology-without-metaphysics">V</a>,
<a href="https://podcasts.ox.ac.uk/2008-lecture-6-revisability-puzzle-revisited">VI</a></li>
<li>Vladimir Voevodsky. <a href="https://www.youtube.com/watch?v=O45LaFsaqMA&t=3118s&ab_channel=InstituteforAdvancedStudy">What if Current Foundations of Mathematics are
Inconsistent?</a></li>
<li>Daniel Sutherland. <a href="https://youtu.be/xXD57a5BEO0">What are Numbers? Philosophy of Mathematics</a></li>
<li>Mark van Atten. <a href="https://www.youtube.com/watch?v=WNAm7TH0iOw&ab_channel=LogicandFoundationsofMathematics">Brouwer and the Mathematics of the Continuum</a></li>
<li>David Corfield. <a href="https://www.youtube.com/watch?v=ijQ5cDL67-0&ab_channel=BrendanLarvor">Good mathematics as good narrative</a></li>
</ul>
</li>
</ul>
<p>Also, there are a lot of interesting articles on logic, philosophy and foundations of mathematics by <a href="https://publish.uwo.ca/~jbell/">John Bell</a>. John was my former mentor. John is a master in tracing back mathematical ideas to their origins in a very rare and particular way which I call “demummification” of mathematical ideas. He has also had a very interesting life which you can read here:
<a href="https://publish.uwo.ca/~jbell/perpmonew4.pdf">Perpetual Motion: The Making of a Mathematical Logician</a>.</p>
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For more serious (and mixed) introductory resources here are some selected papers to get an introduction to some of the most interesting and variegated issues in the philosophy of mathematics:
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<section>
</section>
<h2 id="thematized-resources">Thematized Resources</h2>
<h3 id="on-proofs--proof-assistants">On Proofs & Proof assistants</h3>
<p>Regarding mathematical proofs we shall discuss the following these:
(i) Proofs are abstract mathematical objects (inductively-defined data structures such as plain lists, boxed lists, or trees) which are constructed according to the axioms and rules of inference of the logical system.
(ii) Proof is not manipulation of meaningless syntactic symbols, as the manipulists, or strict formalists, would have it. The knowledge of truth is gained though proof.</p>
<p>We shall argue against the first thesis and discuss our reasons for favouring the second thesis. Our guiding arguments are essentially those of Corcoran’s:</p>
<blockquote>
<p><em>First, belief can be an obstacle to finding a proof because one
of the marks of proof is its ability to resolve doubt. Second,
we usually don’t try to prove propositions we don’t believe or
at least suspect to be true. Third, the attempt to find proof often leads to doubts we otherwise never would have had. If you have a treasured belief you would hate to be without, do not try
to prove it.</em></p>
</blockquote>
<p>However, the second thesis is far from perfect: it does not have the formal force of the first thesis which led to the creation of wonderful mathematical fields such as proof theory and constructive type theory. Also, how does the claim that <q>the knowledge of truth is gained though proof</q> holds against the Zero-Knowledge Proofs (ZKPs) whereby the proof convinces the verifier to any high degree of certinty (strictly below 100%)</p>
<ul>
<li>
<p>Some stuff on proofs</p>
<ul>
<li>Richard Tieszen. What is a proof?</li>
<li>Robert L. Constable. <a href="http://ceur-ws.org/Vol-878/invited1.pdf">Proof Assistants and the Dynamic Nature of Formal Theories</a></li>
<li>Herman Geuvers. <a href="https://www.ias.ac.in/article/fulltext/sadh/034/01/0003-0025">Proof assistants: History, ideas and future</a> Also, look at these <a href="https://www.cs.ru.nl/~herman/ictopen.pdf">slides</a></li>
<li>Hyperproofs</li>
<li>Proof core. The idea is due to the late philosopher David Charles McCarty. See <a href="https://www.youtube.com/watch?v=6uWS7Kwau1A&ab_channel=CopernicusCenterforInterdisciplinaryStudies">What are the limits of mathematical explanation?</a>
Could <em>proof cores</em> (idea engines that propel formal proofs) be one of the conceptual gaps between mathematical proofs and their formalizations? i.e. does a coding of proofs (w.r.t. a formal language) contains strictly less epistemic data/control the proofs themselves? (e.g. (geometric) proof core of -1. -1 = 1, verus one of its proof code in which integers are coded as pairs of natural numbers.) Proof cores are more easily expressible in higher languages, and they are closer to intuition and mathematical cognition.</li>
<li>Robert S.Tragesser, Three insufficiently attended to aspects of most mathematical proofs: phenomenological studies</li>
<li>Scott Viteri, Simon DeDeo, <a href="https://arxiv.org/pdf/2004.00055.pdf">Explosive Proofs of Mathematical Truths</a></li>
</ul>
</li>
<li>
<p>Visual proofs and the role of visual reasoning in mathematics</p>
<ul>
<li><a href="https://plato.stanford.edu/entries/diagrams/">SEP, Diagrams</a></li>
<li>Peirce, C.S., 1933, Collected Papers, Cambridge, MA: Harvard University Press</li>
<li>Barwise, Etchemendy. Visual Information and Valid Reasoning</li>
<li>Allwein, G. and J. Barwise (eds.), 1996, Logical Reasoning with Diagrams, Oxford: Oxford University Press.</li>
<li>Barwise, J. and E. Hammer, 1994, “Diagrams and the Concept of a Logical System”, in Gabbay, D. (ed.), What is a Logical System? New York: Oxford University Press.</li>
<li>Harel, D., 1988, “On Visual Formalisms”, Communications of the ACM, 31(5): 514–530.</li>
<li>De Toffoli, S., 2017, “Chasing The Diagram – The Use of Visualizations in Algebraic Reasoning”, Review of Symbolic Logic, 10 (1): 158–186.</li>
<li>De Toffoli, S. and Giardino, V., 2014, “Forms and Roles of Diagrams in Knot Theory”, Erkenntnis, 79 (4): 829–842.</li>
<li>Halimi, B., 2012, “Diagrams as Sketches”, Synthese, 186(1): 387–409.</li>
<li>Friedman, M., 2012, “Kant on geometry and spatial intuition”, Synthese, 186: 231–255.</li>
<li>John Mumma, Marco Panza. Diagrams in mathematics: history and philosophy.</li>
<li>Roberts, D., 1992, “The Existential Graphs of Charles S. Peirce”, Computer and Math. Applic., (23): 639–663.</li>
</ul>
</li>
</ul>
<h2 id="resources-for-philosophiesfoundations-of-mathematics">Resources For Philosophies/Foundations Of Mathematics</h2>
<p>We will get familiar with the following philosophies and foundations of mathematics.</p>
<h2 id="structuralism"> Structuralism </h2>
<p> The following papers give a good introduction to the structuralist thinking within simple kinds of structures. </p>
<ul>
<li><a href="https://mathcs.clarku.edu/~djoyce/numbers/dedekind.pdf">Richard Dedekind. Was sind und was sollen die Zahlen?</a></li>
<li>
<p>Paul Benacerraf. What numbers could not be. Philosophical Review, 74:47–73, 1965</p>
</li>
<li>Barry Mazur, <a href="http://people.math.harvard.edu/~mazur/preprints/when_is_one.pdf">When is one thing equal to some other thing?</a>, 2007</li>
</ul>
<h3 id="structuralism-and-category-theory">Structuralism and category theory</h3>
<p>A categorical formulation of the structuralist thesis is in</p>
<ul>
<li>
<p>Steve Awodey. “Structure in mathematics and logic: A categorical perspective“. Philosophia Mathematica, 4(3):209–237, 1996</p>
<p>and a follow-up:</p>
</li>
<li>
<p>Steve Awodey. “An Answer to Hellman’s Question: Does Category Theory Provide a Framework for Mathematical Structuralism“, Philosophia Mathematica, 2004</p>
</li>
<li>
<p>Feferman, S. “Categorical Foundations and Foundations of Category Theory”, 1977.</p>
</li>
</ul>
<p>Good introductions to category theory are:</p>
<ul>
<li>Steve Awodey. Category Theory. OUP. 2006</li>
<li>Emily Riehl. <a href="https://math.jhu.edu/~eriehl/context/">Category Theory in Context</a>, Dover, 2016</li>
<li>I. Moerdijk and J. van Oosten. <a href="https://webspace.science.uu.nl/~ooste110/syllabi/toposmoeder.pdf">Topos Theory</a></li>
<li>Brendan Fong and David I. Spivak. Seven Sketches in Compositionality: An Invitation to Applied Category Theory</li>
<li>Elaine Landry. Categories for the Working Philosopher. OUP. 2017</li>
</ul>
<p>On the relationship between the structuralist thesis and the univalent foundation:</p>
<ul>
<li>Steve Awodey. Structuralism, Invariance, and Univalence. Philosophia mathematica 22(1):1–11, 2014.</li>
</ul>
<p>On the relationship between category theory and the set theoretic foundation:</p>
<ul>
<li>Michael A. Shulman. Set theory for category theory</li>
</ul>
<h2 id="formalism">Formalism</h2>
<ul>
<li>Syntax vs semantics: the distinction between the two did not always exist, and it is one of the most fruitful inventions in the history of mathematics.
<ul>
<li>Synthetics vs analtics mathematics: what are the differences and similarities?</li>
<li>Gerhard Heinzmann and Hannes Leitgeb. Formalism, Formalization, Intuition and Understanding in Mathematics: From Informal Practice to Formal Systems and Back Again. Université de Lorraine/CNRS, Nancy 2018.</li>
<li>Jeremy Avigad. Number theory and elementary arithmetic. Philosophia mathematica, 3(11):257–284, 2003.</li>
</ul>
</li>
<li>
<p>Intuitionistic and constructive mathematics and type theory</p>
<ul>
<li>SEP articles: <a href="https://plato.stanford.edu/entries/mathematics-constructive/">Constructive Mathematics</a>, <a href="https://plato.stanford.edu/entries/intuitionistic-logic-development/#BrouLogiGrun">The Development of Intuitionistic Logic</a>, <a href="https://plato.stanford.edu/entries/intuitionism/">Intuitionism in the Philosophy of Mathematics</a>,
<a href="https://plato.stanford.edu/entries/logic-intuitionistic/">Intuitionistic Logic</a></li>
<li>Per Martin-Lof. An Intuitionistic Theory of Types. In G.Sambin and Jan.Smith’s Twenty Five Years of Constructive Type Theory. Oxford Logic Guides Clarendon press, Oxford, 1998.</li>
<li>Per Martin-Lof. Constructive Mathematics and Computer Programming.</li>
<li>Per Martin-Lof. Mathematics of Infinity.</li>
<li><a href="https://github.com/michaelt/martin-lof">Martin-Lof’s collected works</a></li>
<li>Errett Bishop. Schizophrenia of Contemporary Mathematics.</li>
<li>Charles Parsons. The Impredicativity of induction</li>
<li>Anne Sjerp Troelstra and Dirk van Dalen (1988). Constructivism in Mathematics: An Introduction (two volumes), Amsterdam: North Holland</li>
<li>Michael Dummett (1977/2000). Elements of Intuitionism (Oxford Logic Guides, 39), Oxford: Clarendon Press, 1977; 2nd edition, 2000.</li>
<li><a href="http://www2.mathematik.tu-darmstadt.de/~buchholtz/80-518-818/index.html">Ulrik Buchholtz’s course at CMU, Intuitionism and Constructive Mathematics</a></li>
<li>Michael Beeson (1985). Foundations of Constructive Mathematics, Heidelberg: Springer Verlag.</li>
<li>Johan van Benthem. <a href="https://core.ac.uk/download/pdf/191368036.pdf">The information in intuitionistic logic</a></li>
<li>nLab. <a href="https://ncatlab.org/nlab/show/meaning+explanation">Meaning Explanation</a></li>
<li>nLab. <a href="https://ncatlab.org/nlab/show/axiom+of+choice">Axiom of Choice</a></li>
<li>How to implement dependent type theory <a href="http://math.andrej.com/2012/11/08/how-to-implement-dependent-type-theory-i/">I</a>
<a href="http://math.andrej.com/2012/11/11/how-to-implement-dependent-type-theory-ii/">II</a></li>
<li>nLab. <a href="https://ncatlab.org/nlab/show/intuitionistic+mathematics">intuitionistic mathematics</a></li>
<li>Hirschowitz. Intersection types. <a href="https://golem.ph.utexas.edu/category/2008/07/girard_on_the_limitations_of_c.html">Here</a> and <a href="https://golem.ph.utexas.edu/category/2008/07/girard_on_the_limitations_of_c.html">here</a></li>
<li>Bengt Nordstrom, et al. <a href="http://www.cse.chalmers.se/research/group/logic/book/book.pdf">Programming in Martin-Lof’s Type Theory</a></li>
<li>Martín Hötzel Escardó. <a href="https://arxiv.org/ftp/arxiv/papers/1911/1911.00580.pdf">Introduction to Univalent Foundations of Mathematics with Agda</a></li>
<li>Carl Posy. <a href="/files/Phil/phil-maths/posy_brouwer_kant.pdf">Epistemology, Ontology and the Continuum</a></li>
<li>Richard Holton & Huw Price. <a href="http://web.mit.edu/holton/www/pubs/Ramsey.paper.pdf">Ramsey On Saying And Whistling: A Discordant Note</a></li>
<li>Ramsey’s Conception of Theories: An Intuitionistic Approach.</li>
<li>John.L. Bell. <a href="https://publish.uwo.ca/~jbell/PARTS.pdf">Whole and Part in Mathematics</a></li>
<li>John.L. Bell. <a href="https://publish.uwo.ca/~jbell/divvergent.pdf">Divergent conceptions of the continuum in 19th and early 20th century mathematics and philosophy</a></li>
</ul>
</li>
</ul>
<section>
<h2 id="epigraph"></h2>
<div class="epigraph">
<blockquote>
<p> Logic is philosophy’s essence [...] It is the science whereby we are enabled to test reasons. </p>
<footer> C.S. Peirce</footer>
</blockquote>
</div>
</section>
<h2 id="how-to-learn-logic">How To Learn Logic?</h2>
<p><a href="http://classics.mit.edu/Aristotle/prior.html">Prior Analytics, Aristotle, 350 B.C.E
</a>: the first time in history when Logic is scientifically investigated and it is about deductive reasoning, known as syllogism. The Prior Analytics is part of what later Peripatetics called the Organon.</p>
<p>A very good place to learn the basics of modern logic is van Fraassen’s <a href="https://www.princeton.edu/~fraassen/Formal%20Semantics%20and%20Logic.pdf">Formal Semantics and Logic</a>. For other standard references have a look at the <a href="https://www.logicmatters.net/resources/pdfs/Appendix.pdf">big books of logic</a>. Also, for extra resources, see <a href="https://www.logicmatters.net/tyl/">here</a>.</p>
<ul>
<li>
<p>Logic, semantics, metamathematics, Papers from 1923 to 1938, by Alfred Tarski, translated by J. H. Woodger, second edition edited and introduced by John Corcoran, Hackett Publishing Company, Indianapolis 1983</p>
</li>
<li>Logical Paradoxes
<ul>
<li>Hartry Field. Saving Truth From Paradox</li>
</ul>
</li>
<li>Gödel’s Incompleteness Theorems
<ul>
<li>Smullyan’s book</li>
<li>Joyal’s Arithmetic Universes</li>
</ul>
</li>
<li>Tarski’s undefinability (of arithmetical truth) theorem
<ul>
<li><a href="https://en.wikipedia.org/wiki/Tarski%27s_undefinability_theorem">Wikipedia</a></li>
</ul>
</li>
<li>Church’s undecidability theorem
<ul>
<li>Look at Ian Chiswell and Wilfrid Hodges’s Mathematical Logic (OUP, 2007: pp.249)</li>
</ul>
</li>
<li>
<p>Church thesis</p>
</li>
<li>Non-classical logic
<ul>
<li>Priest, G. 2008. An Introduction to Non-Classical Logic: From If to Is, 2nd edition, Cambridge: Cambridge University Press.</li>
<li>Beall, JC, and van Fraassen, B. C. 2003. Possibilities and Paradox, Oxford: Oxford University Press</li>
</ul>
</li>
<li>Modal Logic
<ul>
<li>Belnap, N., M. Perloff, and M. Xu, 2001, Facing the Future, New York: Oxford University Press.</li>
<li>Benthem, J. F. van, 1982, The Logic of Time, Dordrecht: D. Reidel.</li>
<li>Benthem, J. F. van, 2011, Logical Dynamics of Information and Interaction, Cambridge: Cambridge University Press</li>
<li>Blackburn, P., with M. de Rijke and Y. Venema, 2001, Modal Logic, Cambridge: Cambridge University Press.</li>
<li>Chalmers, D., 1996, The Conscious Mind, New York: Oxford University Press.</li>
<li>Goldblatt, R., 1993, Mathematics of Modality, CSLI Lecture Notes #43, Chicago: University of Chicago Press.</li>
<li>Kripke, S., 1963, “Semantical Considerations on Modal Logic,” Acta Philosophica Fennica, 16: 83–94.</li>
<li>Prior, A. N., 1957, Time and Modality, Oxford: Clarendon Press.</li>
</ul>
</li>
<li>
<p>Syntax vs semantics
The distinction between syntax and semantics is one of the fundamental distinctions in the modern logic. We have to understand the origins of this distinction better. Some have traced it back to Frege. Also see this article by <a href="https://cpb-us-w2.wpmucdn.com/campuspress.yale.edu/dist/d/1148/files/2015/10/The-Distinction-between-Semantics-and-Pragmatics-1n5ehuq.pdf">Szabó</a></p>
</li>
<li>Synthetics vs analtics
<ul>
<li>Andrej Bauer. <a href="http://math.andrej.com/2021/02/03/synthetic-mathematics-with-excursion-to-computability/">Synthetic mathematics with an excursion into computability theory</a></li>
</ul>
</li>
<li>Intensional vs extensional</li>
<li>Logical vs non-logical</li>
<li>Logic vs Geometry (sheaves)</li>
<li>
<p>The conception and perception of the universal objects in category theory (c.f. Husserl’s ideas on the universal objects)</p>
</li>
<li>Mathematics as a community activity
<ul>
<li>decay</li>
<li>loss of conceptual understanding</li>
<li>trust is the currency of scientific community</li>
<li>paradigm shift</li>
<li>dogmas</li>
<li>systematization</li>
</ul>
</li>
<li>Current trends and programmes in mathematics
<ul>
<li>The Langlands program</li>
<li>The synthetic methods (category theory and type theory)</li>
<li>Non-commutative geometry and topos theory (appropo Riemann Hypothesis)</li>
<li>P vs NP</li>
<li>Complexity</li>
<li>Pyknotic stuff</li>
</ul>
</li>
</ul>
<section>
<h2 id="epigraph"></h2>
<div class="epigraph">
<blockquote>
We shall not cease from exploration
And at the end of all our exploring
Will be to arrive where we started
And know the place for the first time
</blockquote>
</div>
</section>
<h2 id="semiotics-and-mathematical-notations">Semiotics and Mathematical notations</h2>
<ul>
<li>Louis H. Kauffman. <a href="https://homepages.math.uic.edu/~kauffman/">The Mathematics of Charles Sanders Peirce</a></li>
<li>Cajori, F. 1993. A History of Mathematical Notation, New York: Dover Reprints. (Original published in two volumes by Open Court in London in 1929.)</li>
</ul>
<p>There is a lot to be said about the mathematical notations and symbols: for millenia they have been <em>cognitive technologies</em> enabling us to think previously unthinkable thoughts. For instance, take Khwarizmi’s solution of the problem of solving the quadratic polynomial equation:</p>
<blockquote>
<p><em>If some one says: “You divide ten into two parts: multiply the one by itself; it will be equal to the other taken eighty-one times.” Computation: You say, ten less a thing, multiplied by itself, is a hundred plus a square less twenty things, and this is equal to eighty-one things. Separate the twenty things from a hundred and a square, and add them to eighty-one. It will then be a hundred plus a square, which is equal to a hundred and one roots. Halve the roots; the moiety is fifty and a half. Multiply this by itself, it is two thousand five hundred and fifty and a quarter. Subtract from this one hundred; the remainder is two thousand four hundred and fifty and a quarter. Extract the root from this; it is forty-nine and a half. Subtract this from the moiety of the roots, which is fifty and a half. There remains one, and this is one of the two parts.</em></p>
</blockquote>
<p>And all these words and their meaning are compressed in in modern mathematical notation.</p>
<figure>
<a href="https://en.wikipedia.org/wiki/Muhammad_ibn_Musa_al-Khwarizmi#Algebra" target="_blank">
<br /><img src="/images/posts/2021/jabr-va-moghabele.png" />
</a>
<figcaption></figcaption>
</figure>
<h3 id="and-what-if-i-am-interested-in-set-theory">And what if I am interested in set theory?</h3>
<ul>
<li><a href="https://www.scottaaronson.com/democritus/lec2.html">very basic intro</a></li>
<li><a href="https://ests.files.wordpress.com/2016/01/note1.pdf">THE FIVE WH’S OF SET THEORY (AND A BIT MORE)</a></li>
<li>More resources on <a href="https://github.com/rossant/awesome-math#foundations-of-mathematics">awesome-math</a></li>
</ul>Sina Hazratpourwith focus on the philosophical concerns of the 21st century mathematics