# What is Shinichi Mochizuki's religion

## Philosophical Mathematics I.

Are five apples the same as five pears? Is the group of rotations (in the plane and around the origin) the same as the circle group? Do the two pictures above show the same graphs?

Generally asked: are isomorphic objects the same?

A lot in mathematics revolves around the fact that one wants to see isomorphic objects as the same. Because the addition 3 + 2 is always the same, regardless of whether you add apples, pears or any other object, you can find uniform rules for adding any objects. Because isomorphic groups or graphs are considered to be the same, one can “calculate” with groups or graphs or create lists of them.

“Mathematics is the art of giving different things the same name. ... It is sufficient that these things, although they differ materially, are similar in shape. ”(Henri Poincaré, 1908)

Sometimes it is of course still important to know which objects you are talking about, and not just to know their isomorphism class.

This distinction plays a role in one of the most controversial mathematical debates in recent years, namely that of the correctness of the proof of the abc conjecture. The crucial point in Shinichi Mochizuki's proof is Corollary 3.12, in whose proof, according to Mochizuki, it should be essential to consider certain isomorphic objects as different objects. Jakob Stix and Peter Scholze have found that Mochizuki's arguments become trivial and therefore meaningless as soon as one renounces this distinction between isomorphic objects. However, Mochizuki could not explain or make plausible to them why this distinction between isomorphic objects should suddenly lead to a useful argument, which is why the proof is now viewed by many experts as incorrect or incomplete. (Why abc is still a conjecture.)

A rather bizarre philosophical interpretation of this controversy is now provided by Luboš Motl in an article Category theory as an egalitarian religion.

I think that this philosophy that “isomorphic things must be considered equal” is no longer just a purely mathematical, impersonal, socially neutral meme. It is correlated with some other political and ideological movements that are increasingly ruining the Western societies. Well, look at the statements: "Mathematical objects that are isomorphic must be considered equal." Vs. "All people and their groups - defined by sex, nation, race, sexual orientation, and more - must be considered equal in all circumstances and unequal outcomes must be considered a proof of someone's malice. “The second slogan is clearly an umbrella slogan for identity politics - producing things like“ reverse ”sexism (“ feminism ”),“ reverse ”racism (“ multiculturalism ”), and related pathologies . These pathologies make common sense, ordinary discussions, and rudimentary meritocratic choices increasingly impossible in the West.

But the first slogan is somewhat analogous and it seems rather plausible that its proponents - and proponents of “category theory” - are well aware of this similarity. After all, Roberts' text is titled "A Crisis of Identification"
so aside from the clearly left-wing “equality”, we also have a word with the “ident *” root, something that has an obvious proximity to “identity politics”. What is your identity? Can two isomorphic mathematical objects discussed by a Japanese men accepted to have two different identities, or is it politically incorrect? So it has seemed increasingly likely to me that the likes of Scholze and Stix “don't want” to understand what Močizuki is saying because it conflicts with some ideology that they place above everything else - and the ideology, while completely unjustified, is fundamentally inseparable from the politically ideological delusions of many contemporary Western academics, too.

In this sense, it looks very plausible that “identity politics” may also be blamed for the Westerners' incapability of catching up with the Japanese “arithmetic deformation theory”, a topic that you would normally believe to have zero links with any politics or ideology !

Philosophical interpretations of mathematical facts can sometimes be quite enlightening, but this is just weird.