I am currently reading "Where Mathematics Comes From" by George Lakoff and Rafael E. Núñez - the same Lakoff who authored the seminal "Metaphors We Live By", so I have a lot of time for him. At first it seems like they're just going to explore the pedagogical psychology of mathematics - how interesting! But then right at the end of the preface they hit you with "and by the way this is all there is to it, mathematical Platonism is a lie", which struck me immediately as straying out of their lane. But it seems their investigation into the titular question overwhelmingly led them to this conclusion. The argument is pretty simple - if there is a "platonic mathematics", we cannot have any direct experience of it. All mathematical thought, like all thought in general, is metaphorical. The predictive power of mathematics in the real world is unsurprising because we throw away the metaphors that don't work well.

I do not like this conclusion. Mathematics has always been something of a religion for me. But I can find no flaw with the argument. From a scientific perspective, mathematics bottoms out at "what goes on in human noggins".

I don't think the issue has been as definitively settled as you have been persuaded to think. Let's take a look at the claim "if there is a 'platonic mathematics', we cannot have any direct experience of it" (I realize this is probably a paraphrase of a fuller argument, but it is what I have to work with here.)

Firstly, note the word "direct" here. If it has any relevance, then the authors have assumed the burden of explaining either that there are only direct experiences, or why indirect experiences don't count.

Secondly, what are the premises here? If this is supposed to be axiomatic, then there is literally no reason to either accept or reject it, and claims that the issue has been settled are just statements of belief; otherwise, the argument needs to have premises that are not begging the question in some way. As it stands, this claim is not an argument; it is more of an intuition pump.

Metaphysical discussions tend to (always?) end up as being about the meaning of words like 'real' and 'true'. Whether such discussions can really tell us anything about what must be true is arguably the most meta question in metaphysics.

>The argument is pretty simple - if there is a "platonic mathematics", we cannot have any direct experience of it.

Aside, but this is also Aristotle's exact argument against Platonism in general, though when he makes it in the Nichomachean Ethics he is specifically talking about ethical Good (if the definition/actual taking place of the Good lies in some other plane, we can't participate in it so no one is or can be good), but the idea is the same even when he's talking about what a soul is in De Anima. Aristotle doesn't believe in 'souls' in the way we think of them as religio-spiritual entities that exceed the capacity of the body; a 'soul' for Aristotle is the body but in a way that radically challenges the idea of a body as mere shell or vessel - soul is what any form of life repeats doing, as a body, in order to continue being itself. It should be noted that a lot of time at Aristotle's Academy was spent in Zoology, studying animals and their anatomy.

I'll have to read the book, but in my mind, the (emprical) study of humans and their brains doesn't shed light on the metaphysical question of the nature of mathematics. What they find is how humans have developed to do mathematics. We could have evolved to be the way we are with or without mathematics being "out there". Survival in the physical world would lead us to "throw away the metaphors that don't work well". At any point in time a concrete human being would still be able to consider only a limited set of mathematical ideas i.e. for humans "mathematics bottoms out at "what goes on in human noggins"".

I'd say the patterns you mentioned in an earlier comment are a way for math (or parts of it e.g. some integers) to be "out there". If humans embody mathematics, then analogously so do those patterns.

Ah. I now see this comment too. I think I understand your other statement about "scientifically supported" better. I have also read the book, and I feel it makes a lot of sense. Like I said in my other post: most discourse only acknowledges nominalism or platonism. Neither sat well with me.

Now following Hume and Locke "induction" is often treated as something "invalid", a problem to be solved. If induction is however is not a problem (see for example, Groarke, 2009, An Aristotelian Account of Induction). Aristotelian approaches are reasonable. Hence, numbers and other mathematical concepts can be very real.

Does their theory suggest anything about future AI mathematicians? I'm reminded of the bit in the Cyberiad where Princess Ineffabelle hums a simulated song, and you wonder, is that not a song?