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hiphiphopflipflapflop's avatar

Where do you think the foundations of mathematics come from?

Asked by hiphiphopflipflapflop (6067 points ) March 29th, 2009

Logicism: mathematics is an extension of logic
Formalism: it’s “merely a game” where one picks axioms and applies logic to them to get results.
Intuitionism: it comes from how we as humans perceive time and/or space.
Platonism: we explore a transcendent realm of pure forms.
(there are other schools, and the seriously “undecided” too)

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8 Answers

hiphiphopflipflapflop's avatar

To get it out of the way, I would consider myself one of the “seriously undecided” having been preoccupied by this question for a number of years.

bkudria's avatar

I’d like to point out that the Platonist approach is essentially dualist – we perceive them. To rectify this, there would have to someone exploring our realm (we are transcendent to them). Likewise, the world we’re observing is probably subordinate to a higher reality as well. And, no reason we can’t perceive multiple higher worlds, or be perceived by multiple lower world. So, this would be a Directed Acyclic Graph of many worlds perceiving down the chain.

Harp's avatar

I think that mathematics are the internal ground rules by which the brain organizes relationships. That is to say, pure mathematics doesn’t describe properties intrinsic to external phenomena, but rather describes the terms by which the mind organizes its internal representation of the phenomenal world.

The only argument I have to offer to support this view is the case of Daniel Tammet, the English prodigy who performs truly gigantic mental calculations (including pi to 22,514 places). The reason I think this has a bearing on this question is that Tammet does not do these calculations algorithmically. He doesn’t, in fact, do the calculations; his role is more one of passive observer as the calculations perform themselves in his mental space.

He explains that he experiences number as shape, and that calculations unfold as dynamic transformations of those shapes. This happens without any exertion on his part; he simply reads the resulting shape to arrive at the answer to the calculation.

I can’t reconcile Tammet’s experience with any other explanation than that he is observing, in a far more lucid way than can most of us, the very mechanism of mathematics at work in his own brain.

bkudria's avatar

Having already dismissed (or at least, complicated) Platonism, let me proceed:

Logiscism: Where then, do the foundations of Logic come from? Isn’t Logic just a form of Math?

Formalism: Uninteresting. What’s the point? And why do Humans tend to converge on the same axioms? Is it because be are Human? Would Aliens have a different Math? Completely? Or would they still have basic things, like the concept of a circle? If so, where do the foundations for that “sub-Math” come from?

I guess that addresses Intuitionism too.

Sorry to ask more questions, but this is a really deep topic. Basically, is Math invented or discovered?

hiphiphopflipflapflop's avatar

This fellow wrote in one of his latter books of a desire to infuse richer notions of context into formalism to achieve a “formal functionalism” but I don’t think he went on flesh out that notion.

His followers are absorbed with weak n-categories. I believe they might have something interesting to say on this topic when their theory is ready.

loser's avatar

Me got one rock. You got two rock. We got three rock…

Response moderated
Shuttle128's avatar

Not to plug my own question but I think it’s at least partially relevant, especially in light of Harp’s comment.

I think there are definitely elements of Platonism, however, I think that these elements have a natural origin, and do not exist in a higher plane of existence but in our brain’s neural network structure. If these Forms exist in the structure of our brains it is not far fetched to say that Daniel Tammet can experience these Forms and their combinations directly.

I think that the origins of mathematics are our organization and generalization of observations about the universe. The relationships between things we observe are inherent. We observe relationships and define them. The most rigorous of patterns we define is what we call math.

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