The Nature of Math

[nm420]

Don't know how many people are interested in discussing the philosophy of math, but I didn't want to hijack the Absolute Truth thread with this related, yet sufficiently different, topic. So...

There are two main competing philosophies of math, Platonism and formalism. Platonism charges that the various mathematical objects that are dealt with in the pursuit of math (numbers, groups, fields, etc...) are immutable, eternal and transcendent. They are likened to Plato's Forms (hence the name Platonism), in that they exist independent of human thought and have properties which are immune to our triflings. Formalism maintains that math is purely a human construct, a logical system whose rules we choose and thus is purely dependent on our triflings. An essay I read a few years back claims that most mathematicians are Platonists at heart, but when having to defend their work they will go running to formalism as a justification for the various theorems and whatnot they produce.

The discussion that was started in the Absolute Truth thread hinged on whether mathematical truths can be considered a priori or a posteriori. This is not completely analogous to the two schools of thought mentioned above, though there is another school of thought, call it "historical empiricism" say, which claims that mathematics is an empirical science, as absurd as the notion sounds. There is some evidence for this, however. Throughout the ages, various mathematical "truths" have been found to be untrue. Indeed, this is partly responsible for the birth of formalism as mathematicians of the 19th century grew tired of resting all of their great work on wishy-washy definitions and assumptions. Also, much of the progress of math has been driven by roundly practical needs (though that's not to say most mathematicians care for applying math to the real world). The study of linear equations was prodded along by the need for solutions to distributing sheaths of wheat (or some other such problem faced by administrators of yore). The study of trigonometry was strongly tied to the pursuit of astronomy. Calculus was invented for its practical use in studying physics, so on and so on.

At any rate, what are other's thoughts on this? Is math just a human construct, or are our endeavors merely models of a purer, absolute Truth? Can we say all of our math is completely a priori, uninfluenced by our experiences in the physical world, or is it completely a posteriori, driven by simple observations of our common perceptions? Or is there a mixture of any combination of these, or something else I missed? Discuss at your leisure.

[Plasma Snake[D]]

I think it is a human construct. It is the filter that helps us digest and understand reality. Math helps bridge our understanding with what is. First comes reality, which behaves as it behaves, then comes a human that perceives it and digests it as he does.

[PatrickHenry]

Interesting to me is the base_ten system...A result of having ten fingers?

[tman_ndsu08]

Number systems and math are different.

[Athena]

Interesting question. I wish I were at home where could my books. My first impulse is to agree with Platonism when it comes to math, however, his ideas of perfect models of horses and other things, seem wrong to me. I accept variety egalitarianism. But this is the same for a triangle.

Math is more than numbers, and here is where I need my books to refresh my memory. A triangle is not a number, and it is what it is on earth or Mars or a completely different gallaxy. The power of 3 is physical and the laws of physics do not change, only our understanding of them changes.

Carrying on with Cicero's understanding of God and Natural Law, what is, is, only our understanding of it changes.

[Boetie]

There are two views of the world which is order and chaos. Math only works in the world of order and is useless in the world of chaos.

Chaos would be like an atom disappearing and then reappearing at another location. Which according to some theorists is what actually happens in the subatomic universe. That is why measuring length is impossible.

Take a table for instance how is it possible to put one end of your measuring instrument on an atom at one end of the table and the other end of your instrument on an atom at the other end of the table and keep the atoms in a standstill? you can't, because atoms don't stand still. So measurement is an illusion, which means math can be an illusion as well.

[belverron]

That hardly disproved measurement; it merely questioned its precision at the atomic level.

[weasel]

Math brings power while at the same time confuses everyone.

[Boetie]

belverron posts: That hardly disproved measurement; it merely questioned its precision at the atomic level

The point of my post was to demonstrate that at the atomic level math cannot be exact, it can only estimate or for that matter deal with probability. Math is not an exact science as many believe.

Take for instance Newtonian math, works in everday life but try to apply it to something on the astromonical level, can't be done, thus the theory of relativity is used instead. But the theory of relativity cannot be applied on the atomic level, thus quantum physics is used instead. So you have three different mathematical applications for the whole of reality.

The measurement example was simply to get one the think about whether math is an exact science or not. Recall that I mentioned two worlds chaos and order.

[belverron]

That hardly says, "measurement is an illusion," to me. You might want to use more precise language.

[twoanickel]

Interesting! Long live The Nature of Math thread!

[nm420]

belverron posts: That hardly disproved measurement; it merely questioned its precision at the atomic level

The point of my post was to demonstrate that at the atomic level math cannot be exact, it can only estimate or for that matter deal with probability. Math is not an exact science as many believe.

Take for instance Newtonian math, works in everday life but try to apply it to something on the astromonical level, can't be done, thus the theory of relativity is used instead. But the theory of relativity cannot be applied on the atomic level, thus quantum physics is used instead. So you have three different mathematical applications for the whole of reality.

The measurement example was simply to get one the think about whether math is an exact science or not. Recall that I mentioned two worlds chaos and order.

But shouldn't we differentiate between the practice of pure math and the application of it to real-world models? Newton's calculus is exact (or at least it is now that it lies on an axiomatic system); using it to describe real-world phenomenon isn't, but then humans have yet to discover a means of describing the physical universe infallibly. Likewise, all of the nitty gritty math that quantum physicists use, or those studying cosmology, is real and exact, in the abstract universe anyhow. Using the math as a model, however, is of course inexact, by the very nature of what a model is. Indeed, all of the various "Laws" of physics can generally be boiled down to mathematical statements about the material universe. While the math itself is exact (there is no ambiguity about the statement F=ma), the application of it to physical objects must always be considered a model and thus can never be known to be infallible.