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In the first session we are going to learn about basic questions of philosophy of mathematic from Øystein Linnebo's book "Philosophy of Mathematics".

In the second session we get familiar with the roots of formal logic, formal proofs, and deductivism in mathematics.
Resources:
Note that Linnebo advocates a sort of Fregean Platonism.
Integration Challenge

The challenge is to integrate

• the metaphysics of mathematics (namely, what mathematics is about), with

• its epistemology (namely, how we form our mathematical beliefs).

Integration challenge in basic terms

A Platonist should answer the following pressing question(s):

• "How is it that our ways of forming mathematical beliefs are responsive to what mathematics is about?"

• "How are the practices and mechanisms by which we arrive at our mathematical beliefs conducive to finding out about whatever reality mathematics describes?"

• "Why is it not just a happy accident that our mathematical beliefs tend to be true?" There must be something about what we do that keeps us on the right track.
Linnebo insists that these questions have scientific answers:

"Just as science can be used to study navigation in birds and primates’ knowledge of their environment, it can also be used to investigate human knowledge of mathematics"
And
He holds that the mild circularity does not trivialize the question.

The first philosopher who helps us to have a shot at the "intergration challenge" is Immanuel Kant with his Analytic-Synthetic distinction.

Kant only deals with judgements which are made up of a subject and a predicate, e.g. "Bodies are extended."

They have simple subject-predicate form "A is B."

Kant defines a judgment as analytic if the predicate B belongs to the subject A as something that is (covertly) contained in this concept A.

The predicate expresses nothing which isn’t already contained in the subject.

e.g. "Bodies are extended."

By contrast the statement "bodies have weight" is synthetic.

The predicate amplifies (rather than explicates) the subject.

• Kant agrees with the traditional view that all mathematical knowledge is a priori.

• Naturally, the knowledge of a analytic statement is a priori.

• The truly novel part of his view is that, despite being a priori, mathematical knowledge (including both arithmetic and geometry) is synthetic.
Examples:

• The statement "7+ 5 = 12" is knowable a priori; it’s nevertheless synthetic. Kant's reason:
The concept of twelve is by no means already thought merely by my thinking of that unification of seven and five.

• Consider the statement "A straight line is the shortest path between two points."
The concept of the shortest is therefore altogether additional and cannot be obtained by any analysis of the concept of the straight line.

According to Kant, all synthetic a priori mathematical knowledge is necessary but not directly so; we need a "necessarily present intuition" which supplies the synthetic part of the judgement or statement.

To establish that the shortest line between two points is straight we need to bring in intuition to draw—perhaps in "pure imagination"— the shortest line between two points. We can then perceive that this line is straight.

Problems with the distinction above:

• Since Kant's time, the language of mathematics has developed to include far more complex statements, e.g. statements with bounded and unbounded quantifications, probabilistic statements, etc. Kant's analytic-synthetic distinction does not consider these statements.

• The word "contain" in the definition of analyticity dependes on the underlying logic. Kant's logic is restricted to Aristotle's syllogism which is too restrictive for the purposes of doing any modern mathematics.

Flipping cards

Consider three playing cards A, B, and C , and four admissible operations on them.

• e = do nothing

• α = flip B and C

• β = flip A and C

• γ = flip A and B

Consider the operation * of sequential concatenation on them. e.g. β * α means that first do α and then β.
Strings of bits

Consider strings of bits of lenght two:

e = "00" , α= "10", β= "01", γ= "11"

together with the operation of bitwise XOR on them.
Symmetries of a rectangle which is not a square

Bloch sphere symmetries

Multiplicaiton table

Presentation

Structures and their realizations

"These examples show how completely different systems of objects and relations can instantiate one and the same abstract mathematical structure. This abstract structure has been shown to have two different realizations."
Adopting the viewpoint of structuralism, we should have a language to talk about structures pure and simple.
Category Theory as Mathematics of Mathematics

Category theory is a language of mathematical structures par excellence.
It treats structures as autonomous forms without relying on any specified substance, rejecting the idea that mathematical objects are elements of structured sets.
Category Theory as Mathematics of Mathematics

From the viewpoint of category theory a structure is determined externally, as it were, by its network of mappings to and from other objects of the same kind, rather than internally, in terms of relations and operations on elements.
Category Theory as Mathematics of Mathematics

Category theory is antithetical to the received idea that the meaning of a concept is anchored by reference to a unique absolute universe of sets. Rather it suggests meaning of mathematical concepts depends upon the choice of category of discourse and the meaning varies according to that choice.
Category Theory as Mathematics of Mathematics

Categorical formalizations have more congenial approach to modularity than set-theoretic ones. The idea here is that categories (such as category of monoids, category of groups, category of spaces, etc) are constructed in a certain way to do mathematics in them for a certain purpose. This approach has the advantage that once a result is proved in a category with less structure then one can transport it to categories with more structures by suitable functors.
Category Theory as Mathematics of Mathematics

It even subsumes logic: various fragments of logic correspond to the internal logic of categories.
$G \times G \to G$
The Platonic Heaven of Mathematics

A great number of mathematicians, including many prominent ones, believe in a realm of perfect mathematical entities hovering over the empirical world.

Alain Connes: ‘there exists, independently of the human mind, a raw and immutable mathematical reality, one that is far more permanent than the physical reality that surrounds us’.

Roger Penrose, another unabashed Platonist, holds that the natural world is only a ‘shadow’ of a realm of eternal mathematical forms.

The rationale for this otherwordly view appeared first in the Phaedo. Geometers, Plato observed, talk of perfectly round circles and perfectly straight lines, neither of which are to be found in the sensible world. The same is true of numbers, since they must be composed of perfectly equal units.
The objects studied by mathematicians exist in another world, one that is changeless and transcendent.

In the Meno, Plato explains:
The soul, then, as being immortal, and having been born again many times, and having seen all things that exist, whether in this world or in the world below, has knowledge of them all.

Trouble in Heaven

How mathematicians are supposed to get in touch with this transcendent realm?
Defending the Heaven

Some ways Platonists defend their position:

• Connes believe in some kind of sixth sense: a special sense, ‘irreducible to sight, hearing or touch’, that allows them to perceive Platonic mathematical reality.

• Penrose believes that human consciousness somehow ‘breaks through’ to the Platonic world.

• Linguistic Platonism after Frege.
Kurt Gödel, among the staunchest of 20th-century Platonists, wrote

Despite their remoteness from sense experience, we do have something like a perception of mathematical objects ... I don’t see any reason why we should have less confidence in this kind of perception, i.e. in mathematical intuition, than in sense perception.

Still, a Platonist should answer the following pressing questions:

• How is it that our ways of forming mathematical beliefs are responsive to what mathematics is about?

• How are the practices and mechanisms by which we arrive at our mathematical beliefs conducive to finding out about whatever reality mathematics describes?

• Why is it not just a happy accident that our mathematical beliefs tend to be true? There must be something about what we do that keeps us on the right track.

The Trouble with Conciousness Idea

Mathematicians, like all people, think with their brains, and it’s hard to understand how the brain, a physical entity, could interact with a non-physical reality.

Within the Platonist framework,
we cannot envisage any kind of neural process that could even correspond to the "perception" of a mathematical object.

Mr. Know-it-all comes to the rescue

According to Aristotle, there may be no perfect mathematical entities in our world, but there are plenty of imperfect approximations.

We can draw crude circles and lines on a chalkboard; we can add two apples to three apples, even if they are not identical, and end up with five. By abstracting from our experience of ordinary perceptible things, we arrive at basic mathematical intuitions, and logical deduction does the rest.

The Trouble With Aristotle's Down-to-Earth Approach

There is one putative mathematical object that the Aristotelian view can’t handle: infinity. We have no experience of the infinite.

An actual ‘completed’ infinity, as opposed to a merely potential one, is something we never encounter in the natural world.

Infinite Ban

For Eleatic philosophers, the idea of infinity was long regarded with suspicion, if not horror.

Zeno’s paradoxes seemed to show that if space/time could be divided up into infinitesimal segments, then motion would be impossible.

This absurd conclusion led Aristotle to ban infinity from Greek thought.

The Infinite Monster Rearing Its Ugly Head

Already with the Newton’s and Leibniz’s invention of calculus and the use of infinitesimal, actual infinities are reintroduced into mathematics. Aristotlean view no longer secures a philosophical certainty about mathematics.

Cantor Bursting Onto The Scene

It was in the late 19th century that Georg Cantor, a Russian-born German mathematician, supplied the theory needed.

Cantor did not set out to explore infinity for its own sake; rather, he claimed, the task ‘was logically forced upon me, almost against my will’.

What he ended up with, after two decades of intellectual struggle, was a succession of higher and higher infinities – an infinite hierarchy of them, ascending towards an unknowable terminus that he called the Absolute.

The Pursuit of Absolute

According to Dauben, the pursuit of "Absolute" seemed to him a divinely vouchsafed vision; in transmitting it to the world, he regarded himself as ‘God’s ambassador’.

Cantor spent the rest of his life pondering the theological implications of infinity. He ended his life in an asylum.

Cantor's Reception in Europe

The Reception of Cantor's new theory was particularly terrible in France. Henri Poincaré (who rivalled Hilbert as the greatest mathematician of the era) observed that higher infinities ‘have a whiff of form without matter, which is repugnant to the French spirit’.

Russian mathematicians, by contrast, embraced the newly revealed hierarchy of infinities.

In Germany the Reception was divided: His one-time teacher Leopold Kronecker reviled it as ‘humbug’ and ‘mathematical insanity’, whereas David Hilbert declared: ‘No one shall expel us from the paradise that Cantor has created for us.’

French Rationalism vs Russinan Mysticism

Why the contrary reactions? Loren Graham and Jean-Michel Kantor see it as a matter of French rationalism versus Russian mysticism.

Name Worshippers

The founding figures of the most influential school of 20th-century Russian mathematics were members of a heretical religious sect called the Name Worshippers, who believed that by repetitiously chanting God’s name they could achieve fusion with the divine.

Name Worshipping, traceable to fourth-century Christian hermits in the deserts of Palestine, was revived in the modern era by a Russian monk called Ilarion. In 1907, Ilarion published On the Mountains of the Caucasus, a book that described the ecstatic experiences he induced in himself while chanting the names of Christ and God over and over again until his breathing and heartbeat were in tune with the words.