Thanks to this discussion I now have a new idea that popped into my head:

Is mathematics an emergent phenomenon?

Hear me out on this one. Obviously we have to distinguish between a) that which mathematics describes and b) the formalism itself. But I think that in some cases an argument could be made that this line is blurred a bit. In any case, as we go “down the food chain,” so-to-speak, the understanding of mathematics that species have most likely decreases (for example, I would be shocked if ants could figure out a square root). Presumably the phenomena described by the mathematics is still there though. Where does complex mathematics “emerge” from this (if it emerges at all – perhaps it’s simply “there”)?

I don’t know why the nature of mathematics is so on my mind lately, but there you have it.

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I think that inasmuch as mathematical ideas have psychological meaning, they are emergent.

Mathematical formalism can be thought of as simple symbol shuffling, and mathematical truths can be thought of as the logical results of axioms and operations, but most people (meaning me) don’t simply shuffle symbols around to discover meaningless consequences of arbitrary assumptions. Instead we impose meaning on the symbols and use our intuition with regard to that meaning to guide our symbol shuffling. This is especially obvious in the mathematical sciences where many of the symbols directly represent physical objects or processes, but I believe it also applies to pure mathematicians whose work is more abstract and has no direct application to anything practical (yet). Math is largely a creative endeavor, and to be any good at it one must have some insight into the ‘behavior’ of the symbols and the ideas they represent. Mathematical ideas come to have meaning to the person manipulating them, even if that meaning can’t be expressed in everyday language. In fact, to the inventor (or discoverer) of a certain mathematical idea, the meaning often arrives before the formalism, instead of the other way around.

I would say that the human tendency to find meaning in mathematics is emergent, just like the rest of human personality and psychology (though I’m sure there are plenty of philosophers who would disagree with that). Since there is no clear bridge between physics and consciousness, I think of consciousness and ‘meaning-making’ as an emergent property of our physical brains. We can give physical descriptions of how neurons fire (sodium-potassium pumps and all that), but we can’t explain precisely how large numbers of these neurons can work together to generate visual images, memories, preferences for certain flavors of ice cream, etc. At best we can make associations between types of mental experience and activity in certain regions of the brain, but the mechanism of consciousness does not seem amenable to mechanical explanation.

On the other hand, it is easy (and probably unverifiable) to imagine how the tendency of humans to think mathematically evolved. The ability to count or at least to understand ‘greater than’ and ‘less than’ helped early humans (and other animals) to stockpile food and to avoid fighting when overmatched. The ability to visualize parabolas is helpful when you want to hit something (food or an enemy) with a rock or spear. And so on.

This idea leads to the thought that mathematics is less universal than we often suppose. That is, there can be many different mathematical systems that are not necessarily consistent with each other. The ‘facts’ of mathematics depend on which axioms and operations you choose (think of Euclidean vs. non-Euclidean geometry). The mathematics that evolves out of another species on another planet would probably be initially tailored to the survival of that species. What kind of mathematical ideas would be useful to, say, some kind of intelligent plasma? Counting might not be the first thing they come up with. What about dolphins? They are highly intelligent, but their primary sense is hearing, not vision, and they don’t have fingers to count with. Certainly if they adopted our axioms and operations, they would necessarily be led to the same conclusions as us, but who’s to say their different perspective wouldn’t be lead them off in some other mathematical direction that has no parallel with what we commonly accept as universally true mathematics?

Those are excellent points. It leads to the question of whether mathematics is actually nothing but formalism.