Resources:

- Øystein Linnebo. Philosophy of Mathematics, Princeton University Press, 2017
- Aristotle. Prior Analytics, circa 350 BCE, translated by Gisela Striker, Oxford: Clarendon Press 2009
- An older and mainstream translation: Robin Smith, Indianapolis: Hackett, 1989.

Note that Linnebo advocates a sort of Fregean Platonism.

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.

"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.

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.

"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 is a language of mathematical structures par excellence.

It treats structures as

From the viewpoint of category theory a structure is determined

Category theory is antithetical to the received idea that the

Categorical formalizations have more congenial approach to

It even subsumes logic: various fragments of logic correspond to the

$G \times G \to G$

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.

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.

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

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.

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.’

Why the contrary reactions? Loren Graham and Jean-Michel Kantor see it as a matter of

The founding figures of the most influential school of 20th-century Russian mathematics were members of a heretical religious sect called the

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.