What is Mathematics?

[Elena]

Joshua, having not read Badiou (yet), I will reply only to your interpretation of his response to the critical starting point in philosophy, which is indeed compelling. I think in the end I will stick to my guns, perhaps out of existential humility (or timidity), and stick to the critical approach, if only because I am keenly aware of human fallibility - including and especially our proneness to fall prey to our own cognitive projections. Not all people share this overriding motive with its accompanying intuition, and therefore their thought is wrested in altogether different directions. We are therefore in the end left with your very pertinent question as to where the directing motive and decision come from.

I would agree with the implication of your interpretation of Badiou's smart move, which seems to carry with it an implicit critique of that great hole in Kant's argument which I've touched on elsewhere - namely, his inability to articulate the ontology of thought. Obviously thought, too, as well as the phenomenological domain, have some kind of ontological being, but what? And it is here that we're left in the dark. So I agree when you say that: "What I appreciate about this attempt is that an effort is made to locate and deploy our capacity for abstract thought and not simply to confine it to an elevated but rather precious space." I appreciate such an effort, too. But when you next go on to implicitly question human finitude by virtue of our (debatable) ability to conceptualize infinity, you veer towards the danger that the critical POV strove to avert us from. Certainly, the innoculation from illusion that the critical approach provides as well as the safety it brings comes at the price of humbling us, and certainly there is a "foreclosure" at play in the approach as we stick to ground we are sure reason actually possesses within itself instead of prematurely “extending it” through positing cognitive artifacts that we then come to pretend are discovered "out there."

But again, the problem I have with the realism that is offered as an alternative (even a qualified realism such as Badiou's here) is that it simply choses to ignore the all-pervasive conditioning of our access to reality effected by our cognitive apparatus. Realism still holds to the ancient dogma of mind as mirror of nature. How can this be, after advances in cognitive psychology that insist mind is constructive through and through? There will always be some mathematician or physicist who will come up and glibly insist that “Yup, mathematics is a way to describe nature,” but I doubt that such folks understand the cognitive science that studies the (constructive) nature of description, of modeling, as well as the species-specific (because adaptive) selectivity of abstraction and symbolic representation in general.

Which brings me to a critical issue related to our topic: which science you regard as primary, whether physics or a human science like cognitive science, will affect your epistemology and therefore your ontology. Physicists are paradigmatically realists and always have been; psychologists and cognitive scientists are often constructivists and their ontology consists of paying lip-service to materialism while practicing some form of idealism. Now where does that leave math and logic? It depends who you ask, the cognitive scientist or evolutionary psychologist who studies the apparatus that makes possible math/logic and the way it is situated within the context of our species' evolution (and is therefore justified only pragmatically), or the physicist who uses this apparatus with the confidence that it opens up reality before him, and that it has a mindish objective correlate in nature (realism). Badiou's solution (as interpreted by you), if it can lead to denying the ultimate context of our biological finitude, is unsatisfying, to say the least. At least Kant was realistic and true enough to our nature to make finitude the fundamental fact of our being, and to acknowledge the primacy of our (biological, species-specific) form in dealing with these questions. That is, to me, the most sensible context within which all thought must be understood. The alternative is mind treated as some kind of disembodied haze, unconstrained by the pressure of its formal situatedness and therefore a prey to its own illusions (or the parody of rationality that Kant discusses when he talks about Plato's dove). I see from my brief Googling of Badiou that he did indeed attempt to repackage Platonism for an ultra-refined postmodern audience.

[Richard]

Hi Everybody,

My name is Richard and I have just started a YouTube channel on Classics, focusing on Sanskrit. Please check out my first video, a short trailer, https://youtu.be/2Pkhwd6xX68

We are working on a piece concerning Indian mathematics composed in Sanskrit, much of the content may not have been easily accessible in English before. It will be out later this year.

[Feliks]

Nice thread to revive.

This catches my eye:

Brian wrote: "I'll break it down this way; if there were aliens who came to earth after humans completed the annihilation of their species from war or disease or something, would the aliens be able to understand our math or better yet, would math exist for them, or even at all? I want to say no. No because it is very possible that the mechanical system they used to build a spaceship and navigate to our Earth was based on something which was intrinsic to their perception of reality. ...."

Let's state this more conservatively. Granting a 'different' perception of reality to a creature is not the same thing as allowing that they might have 'no' perception of some reality which we perceive. I would treat either case in it's own respect, rather than mixing them together.

Brian wrote: "Maybe they do not understand reality in a dimension equivalent to ours and mathematics is not exactly a fundamental principle for generating the proper mechanics for space travel. ...."

Sure, that could be, anything could be possible 'in a different dimension'; but that's leapfrogging over what we can more safely (more firmly), dare to conject.

Brian wrote: "For instance do honeybees or ants need mathematic complete their objectives? Maybe we can see a mathematical pattern in what they are constructing and trying to achieve but that is due to or own necessity of seeing the world mathematically...."

How can you say so certainly that we see mathematics in the world simply due to our psychology? Anyway, for data on the math behind bees and ants, refer to D'arcy Wentworth Thompson. The engineering data is there if anyone is interested.

[Feliks]

I'd like to try to answer myself the objection I raised two posts above.

Brian wrote: "I'll break it down this way; if there were aliens who came to earth after humans completed the annihilation of their species from war or disease or something, would the aliens be able to understand our math or better yet, would math exist for them, or even at all? I want to say no..."

What would our conjecture be here, if we were speaking not of 'mathematics', but of the concept of the 'soul'? When we encounter 'other creatures' (other species of creature on earth for example, or other cultures) don't we admit that no matter whether they 'understand' the concept ('soul') or not, they still possess a soul regardless? This is the appraisal towards which any coherent religion usually arrives.

Thus, aren't 'aliens' --no matter what dimension they inhabit --either 'living' or 'non-living'? Do not all living creatures possess a metaphysical aspect that surpasses their mental capacity to identify it? We never rule that any living creature is without spiritual energy. There is no organism so low or so strange that we could say it does not. Even plants feel pain. In the same way, nothing in the material universe can escape the scope of mathematics simply by failing to mentally recognize numerals. 'Universality' is not strictly a human bias.

Mathematical principles must underpin anything corporeal. Yes, an organism without the sense to detect 'music' might not grasp the idea of a 'song' --but there are other principles which it still must align with.

My opinion is that it should not ever matter (to the question of 'what is mathematics?) whether we encounter aliens who don't understand mathematical concepts the same way we do.