NancyLebovitz comments on The role of mathematical truths - Less Wrong

14 Post author: SilasBarta 24 April 2010 04:59PM

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Comment author: Jack 25 April 2010 02:14:42AM *  7 points [-]

That there is an immaterial realm of ideal forms (structures, concepts) of which our universe consists solely of imperfect approximations of.

This stuff about imperfect approximations is just a remnant of Plato's mysticism. Few modern platonists would say anything like that. This notion of an immaterial "realm" has similar connotations. How about:

Platonism is the view that there exist such things as abstract objects — where an abstract object is an object that does not exist in space or time and which is therefore entirely non-physical and non-mental.

Platonism is appealing because it adheres to our norm of accepting the existence of things we make true statements about. "Silas is cool" implies the existence of Silas. Similarly, "3 is prime" implies the existence of 3. The list of non-platonist options as far as I can recall consists of: mathematical objects are mental objects, mathematical objects are physical objects, statements about mathematical objects are false (like statements about Santa Claus), or statements about mathematical objects are actually paraphrases of sentences that don't commit us to the existence of abstract objects.

It seems like you are trying something like the last. But for this strategy you really should give explicit paraphrases or, ideally, a method for paraphrasing all mathematical truths.

I would say instead that there is some generating function for reality. A system of inscriptions/rules can describe that generating function imperfectly; but this in no way means that the rule/inscription system has some existence apart from its instantiation as the universe itself, and again explicitly in a model.

But then what kind of thing is this function? It clearly isn't merely a set of inscriptions and rules for manipulating them (the models). Nor is it merely the physical universe. We talk like it exists. If it doesn't, why do we talk like this and what do claims about it really mean?

Comment author: NancyLebovitz 25 April 2010 09:33:34AM *  2 points [-]

At least for geometrical forms, the abstractions may be intrinsic to the mind, even if they don't exist outside it.

In The Man Who Mistook His Wife for a Hat, there's a description of a man who lost the ability to visually recognize ordinary objects, though he could still see. The one description suggest that he just saw geometry.

In Crashing Through, which is about a man who lost his sight at age 3 and recovered it in middle age and which has a lot about recovered vision and the amount of processing it takes to make sense of what you see, there's mention of some people who are very disappointed when they recover their sight-- they're constantly comparing the world to an idea of it which is perfectly clean and geometrical.

Comment author: Jack 25 April 2010 10:47:38AM *  0 points [-]

In <i>The Man Who Mistook His Wife for a Hat</i>, there's a description of a man who lost the ability to visually recognize ordinary objects, though he could still see. The one description suggest that he just saw geometry.

I'm a little confused: did is visual field lose focus such that, instead of seeing the details on objects and their imperfections he actually just saw idealized geometric figures?

One problem with this as evidence of the possibility that geometric forms could exist only in the human mind is that it presumably only applies to a rather narrow class of geometric forms. It would be weird if the geometric forms we have innate access to had a different ontological status from forms that can't be instantiated in the human mind: like a 1000-sided polygon or something in 4+ dimensions.

Comment author: NancyLebovitz 25 April 2010 11:04:50AM 3 points [-]

What I meant was that, if people have simple geometric forms built deep into their minds, then it would be tempting to conclude that math has an objective eternal existence because it feels that way.

In any case, I found the actual quote, and I've very uncertain that it suggests what I thought it did. It seems as though the man was at least as sensitive to simple topology as geometry, However, people don't romanticize topology.

Here's the passage, which I had not remembered as well as I thought:

I had stopped at a florist on my way to his apartment and bought myself an extravagant red rose for my buttonhole. Now I removed this and handed it to him. He took it like a botanist or morphologist given a specimen, not like person given a flower.

“About six inches in length,’ he commented. ‘A convoluted red form with a linear green attachment.’

‘Yes,’ I said encouragingly, ‘and what do you think it is, Dr P.?’

‘Not easy to say.’ He seemed perplexed. ‘It lacks the simple symmetry of the Platonic solids, although it may have a higher symmetry of its own… I think this could be an inflorescence or flower.’

‘Could be?’ I queried.

‘Could be,’ he confirmed.

‘Smell it,’ I suggested, and he again looked somewhat puzzled, as if I had asked him to smell a higher symmetry. But he complied courteously, and took it to his nose. Now, suddenly, he came to life.

‘Beautiful!’ he exclaimed. ‘An early rose. What a heavenly smell!’ He started to hum ‘Die Rose, die Lillie…’ Reality, it seemed, might by conveyed by smell, not by sight.

I tried one final test. It was still a cold day, in early spring, and I had thrown my coat and gloves on the sofa.

‘What is this?’ I asked, holding up a glove.

‘May I examine it?’ he asked, and, taking it from me, he proceeded to examine it as he had examined the geometrical shapes.

‘A continuous surface,’ he announced at last, ‘infolded on itself. It appears to have’ – he hesitated – ‘five outpouchings, if this is the word.’

Comment author: Jack 25 April 2010 11:36:02AM 1 point [-]

It's a wonderful extract in any case. It is fascinating to see someone describing the world without anything more than the phenomenology of his surroundings. It is interesting that the concepts he had access to were mathematical and geometric- that these concepts involve a part of the brain separate from the part that involves more complex and obviously learned concepts like shoe, glove, and flower does seem important to keep in mind when evaluating the evidence on this issue. You're right that this fact could lead to us positing a false ontological difference... though of course there are those who will say "gloveness" and "flowerness" are abstract objects as well. The fact that these concepts are processed in different parts of the brain could also be taken as evidence for the distinction in that different evolutionary processes generated these two kinds of concepts. I'm not sure how to interpret this. Good for keeping in mind though.

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