James Franklin has an interesting piece today at Aeon, asking what exactly mathematics is. He looks at Nominalism and Platonism, but discounts both in favor of Aristotelian Realism, which is something I’d not heard of before but seems equivalent to the idea that mathematics is empirical.

What is mathematics about? We know what biology is about; it’s about living things. Or more exactly, the living aspects of living things – the motion of a cat thrown out of a window is a matter for physics, but its physiology is a topic for biology. Oceanography is about oceans; sociology is about human behaviour in the mass, long-term; and so on. When all the sciences and their subject matters are laid out, is there any aspect of reality left over for mathematics to be about? That is the basic question in the philosophy of mathematics.

via What is mathematics about? – James Franklin – Aeon.

I’m in the final chapters of Max Tegmark’s ‘Our Mathematical Universe’ and will have a lot of thoughts about this soon. Right now, I’ll just note that I do happen to think we learn mathematics initially through perception (some of which is instinctual) but that many derived mathematical structures built on top of those empirical foundations are non-physical.

The question is are those non-physical concepts “real”, meaning Platonism, or are they simply structures built on an incomplete or simplified understanding of our reality, that is built on quantitative relationships and patterns we have observed, but ignoring or in ignorance of constraints that prevent the physical embodiment of those derived structures?

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Hi SAP,

Perhaps unsurprisingly, I don’t find his arguments convincing.

As you know, I’m for Platonism, so I will concentrating on critiquing his account of Platonism.

Firstly, I resent the mystical terms he uses to describe Platonism. This is a stereotype I wish we could do away with.

Mathematical objects do not “float somewhere outside our own universe of space and time”. They do not exist in a “mysterious realm of forms”. There is no such realm and there is certainly no floating going on. Mathematical objects exist, but they do not exist in a place, and it is wrong to think of them this way. They are not physical so the concept of floating does not apply. There is nothing mysterious about them. We can all intuitively grasp the sense in which they exist, we do so all the time when we use the verb “to be” about mathematical objects, e.g. “there are infinitely many primes”, “there is no greatest integer”.

Platonism seems mysterious and frankly absurd only when painted in this light. These terms may seem to be only poetic figures of speech, but they are a slander on Platonism in my view.

“it’s hard to imagine how can we see them or have any other perceptual contact with them.”

We don’t. We infer them.

” Five-year-olds learning to count don’t perform sophisticated inferences about abstractions”

Five-year-olds are not blank slates. They are evolved creatures and are equipped with some basic computing hardware that has an intuitive grasp of the fundamentals of mathematics (e.g. 2 is more than 1). They don’t, however, infer the existence of abstract mathematical objects. Five year olds are not Platonists. Few of them give much thought to the philosophy of mathematics.

“Such properties are not abstracta: like colours, they have causal powers that result in our seeing them.”

I don’t see how this is an argument against Platonism. Physical properties map onto abstract properties in a way which preserves the relations between them (isomorphism). But those abstract truths hold whether or not there is a physical manifestation of those truths. So the abstracta exist independently of any isomorphic physical instantiation.

The argument proposed for Aristotelian realism seems rather weak. It explains that many mathematical properties are represented in the world, but fails to give much of an account of why there is so much to discover about mathematical properties that have no known physical significance or any of the other motivations for Platonism, e.g. how it is that two mathematicians can independently discover the same truths.

I don’t know for sure, but the author seems to be motivated by some kind of supernaturalism:

“The scholastics, the Aristotelian Catholic philosophers of the Middle Ages, were so impressed with the mind’s grasp of necessary truths as to conclude that the intellect was immaterial and immortal. If today’s naturalists do not wish to agree with that, there is a challenge for them.”

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Hi DM,

I figured you’d have an opinion on this. I’ll agree that his language toward Platonism seems biased. He doesn’t seem to seriously consider it or give it a fair hearing. I also noticed the shot at naturalism which seemed like a gratuitous indulgence, and a risky one since all it will take is a computer program unearthing new mathematical constructs to invalidate it.

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We don’t say colours are Platonic entities (whatever mappings there may be), because we see them, and we see them because they causally affect us. It’s the same with (some) mathematical properties like ratios: I (and five year olds) can see the ratio of my height to yours (if we stand near each other). There’s no need for abstracta (in any sense) in that.

How to discover things about mathematical properties that aren’t physically realised is a fair question: the answer is in principle the same as how to know about an uninstantiated shade of blue. More in the book, An Aristotelian Realist Theory of Mathematics.

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I don’t think colours are a good example.

I would say colours are not even abstracta, they are qualia. Colours as we know them don’t exist in any objective sense. We perceive light at 580 nm (yellow) to have the same colour as an equal parts mixture of light at 680 nm (red) and 510 nm (green). The colour perception of different people can also vary. Some people are colour blind, and there is some evidence that some people can see into the ultra-violet spectrum to some degree. There is no wavelength of light that corresponds to magenta. The wavelengths themselves are fundamentally ratios and so not really any different from the ratio of two heights.

The perception of audio pitch has more of a direct (albeit logarithmic) correlation with an objective physical phenomenon, but I would still distinguish between the qualia of pitch and the ratio of frequency. The frequency is not perceived directly but automatically mapped to pitch which is what we perceive.

The ratio of heights is not perceived directly but inferred from sensory data. 5 year olds can do this automatically because they have the hardware to do so, just like a computer vision system could do it without needing any beliefs about the philosophy of mathematics. The concept of abstracta is needed not for working with ratios but for understanding and describing what is happening when we do so.

So ratios do causally affect us, all the time, as do even more abstract mathematical ideas such as the Mandelbrot set (it has caused me to refer to it here, after all). I just don’t see how this can be construed as an argument against their independent existence. It only means (a) that we have the ability to infer their existence and (b) that there are physical phenomena which share isomorphisms with abstracta and so which behave according to the same rules as the abstracta, e.g. physical triangles obeying Pythagoras’s theorem.

Good luck with the book! Will it be available on Kindle?

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Dr. Franklin, appreciate you stopping by. I had missed that there was a book. I second DM’s well wishes on its success and interest in a Kindle edition.

I see where you’re going with the uninstantiated shade of blue example, but along with DM’s objection, I’m struggling to think of a more objective example, like maybe a physical phenomena that is possible under the laws of nature but hasn’t been observed anywhere, like maybe an antimatter star? Again, I’m left with wondering if such unrealized objects are “real” are just conceptual mistakes based on our lack of knowledge of all the constraints.

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Five year olds aren’t very useful in this discussion, because they are sophisticated mathematicians with a lot of hands-on experience of the world. More interesting is the baby aged between about 6 months and 12 months, beginning to understand the world. I sat amazed watching my son (who 12 years later was awarded his school’s mathematics prize) grappling with the concept of size. Day after day, week after week, month after month, he would attempt to fit the large plastic beaker inside the small plastic beaker. He did not give up with this project until he had learned an empirical fact about the world – that some things are bigger than others.

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This is somewhat why I think the foundations of mathematics and logic are empirical, with some help from evolved instinct. It doesn’t feel empirical because it’s both among the earliest things we ever learned about the world, and because some aspects of it are pre-wired into our minds.

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I agree that many of the mathematical abilities of 5 year olds are learned. But I think Dr Franklin’s point was that they have these abilities without needing to consciously adopt mathematical realism.

Of course this is beside the point because the philosophy of mathematics is a separate concern from the application of mathematics.

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