Zigmen wrote:
Is math a science? Or is it a tool?

Good question!

Richard Feynman once flippantly declared that "Mathematics is to physics what masturbation is to sex."

Whether mathematics is the servant of physics or its master is, I think, largely in the eye of the beholder.

As a discipline unto itself, mathematics is a science rather unlike other sciences. Whereas the physical science of Ptolemy's Almagest has been nearly entirely supplanted by better theories, the theorems of Euclid's Elements have been eternal. The physical sciences are investigative, relying on experiment and trial and error; mathematics are based on an underpinning of logic; what was proven logically 2000 years ago is just as true today. Science has no luxury of proof. A scientific theory and a mathematical theorem are fundamentally different animals.

Mathematics builds a wall on the undisturbed foundation of previous mathematics; science takes a wrecking ball to its wall periodically in order to rebuild it.

sweitzen wrote:what was proven logically 2000 years ago is just as true today.

This seems dubious to me. 2000 years ago, I suspect that mathematicians felt that they were dealing with truths about reality, whether Platonic or not. I cant think of any case before Cardano, in which mathematics was at odds with observation, and it wasn't until the twentieth century that formalist or fictionalist positions became established. But now they are established, and we have examples of statements which are true in some formalisms but false in others, so it's now difficult to talk about the truth of mathematics in a way that would've made sense 2000 years ago.

Example?

Someone wrote:Example?

In Martin-Lof type theory the axiom of choice is a theorem, in van Lambalgen's ZFR, the axiom of choice is provably false.

ughaibu wrote:This seems dubious to me. 2000 years ago, I suspect that mathematicians felt that they were dealing with truths about reality, whether Platonic or not. I cant think of any case before Cardano, in which mathematics was at odds with observation, and it wasn't until the twentieth century that formalist or fictionalist positions became established. But now they are established, and we have examples of statements which are true in some formalisms but false in others, so it's now difficult to talk about the truth of mathematics in a way that would've made sense 2000 years ago.

I'm coming at this from a physicist's perspective, looking at the central difference between mathematics and observational, experimental science, so perhaps some of the subtlety is lost on me. I had in mind, for example, Euclidean geometry. The theorems and proofs of Euclid's time, 2000+ years ago, are still true today. Relaxing the parallel postulate gave us non-Euclidean geometries, true, but in any Euclidean space where the parallel postulate holds true, the results of Euclid are valid -- newer mathematics have not proven them wrong.

I've had Morris Kline's _Mathematics: The Loss of Certainty_ on my wish list for a while. I suspect it deals with this issue exactly -- perhaps I should go get it now.

sweitzen wrote:I've had Morris Kline's _Mathematics: The Loss of Certainty_ on my wish list for a while. I suspect it deals with this issue exactly -- perhaps I should go get it now.

It's a very boring book and mainly a history of maths. Stanford has various articles giving overviews of the different systems and positions: http://plato.stanford.edu/entries/philo ... thematics/

ughaibu: But wouldn't you admit that such statements don't actually contradict one another? It's similar to the fact that 1+1=0 if you're working in modulo 2 arithmetic. The truth of all mathematical statements is always dependent on context. It's a mark of a change in the degree of complexity of the mathematical enterprise that we're dependent on context in this way, not a real change in the nature of mathematical truth per se. If anything, we are more close to absolute truth than in the classical period. Recall the attempts to prove the Parallel Postulate from the other axioms of geometry.

Someone wrote:. . . . wouldn't you admit that such statements don't actually contradict one another?

How would you answer the question of whether or not the axiom of choice is true?

Someone wrote:If anything, we are more close to absolute truth than in the classical period.

If there are different mathematical systems in which incompatible results can be proved, either it is the case that mathematical correctness doesn't indicate truth, in a correspondence sense, or it's the case that there exists an "actual" mathematics which is independent of mathematising agents. In the latter case, the same considerations apply to logic, so there is no non-circular way to attempt to identify the "actual" mathematics other than by observation. It seems to me that the Greeks based their mathematics on observation, and they considered mathematical truth to be truth by correspondence. Modern mathematics is full of stuff which is meaningless in terms of observables, or is in direct conflict with observation, so I dont see how the notion of "absolute truth", about mathematics, can be defended.

ughaibu wrote:It's a very boring book and mainly a history of maths. Stanford has various articles giving overviews of the different systems and positions: http://plato.stanford.edu/entries/philo ... thematics/

Thanks!

(And there goes my productivity for the day. That is what we call a successful nerd snipe.)

ughaibu wrote:

Someone wrote:. . . . wouldn't you admit that such statements don't actually contradict one another?

How would you answer the question of whether or not the axiom of choice is true?

Someone wrote:If anything, we are more close to absolute truth than in the classical period.

If there are different mathematical systems in which incompatible results can be proved, either it is the case that mathematical correctness doesn't indicate truth, in a correspondence sense, or it's the case that there exists an "actual" mathematics which is independent of mathematising agents. In the latter case, the same considerations apply to logic, so there is no non-circular way to attempt to identify the "actual" mathematics other than by observation. It seems to me that the Greeks based their mathematics on observation, and they considered mathematical truth to be truth by correspondence. Modern mathematics is full of stuff which is meaningless in terms of observables, or is in direct conflict with observation, so I dont see how the notion of "absolute truth", about mathematics, can be defended.

I'm sorry that I only have a few minutes per day for this, but my position is pretty much formalistic. If you have consistent axioms (in whatever isolated setting), then the things that are provably true are the ones you get to through some chain of reasoning from them. You can stipulate an axiom or its negation, but within the particular setting what's going on in settings with the opposite choice is not relevant. Generally, I have not known of a useful system that says the axiom of choice is false, in the normal sense of the word 'useful', but I may not be up-to-date. I understand from perusing a recent American Mathematical Monthly article, that there is actually some argument now about inconsistent systems' usefulness. So, I might a bit off. That seems bizarre.

Someone wrote:If you have consistent axioms (in whatever isolated setting), then the things that are provably true are the ones you get to through some chain of reasoning from them.

Fine, but whatever you mean by "truth" here, it isn't correspondence theory truth. Accordingly, you appear to be agreeing with me, unless you think that the Greeks, 2000 years ago, similarly viewed mathematical truths under a formalist, rather than a correspondence theory.

Someone wrote:I understand from perusing a recent American Mathematical Monthly article, that there is actually some argument now about inconsistent systems' usefulness.

Melbourne University is presently hosting a three year project on the extention of para-consistent set theory. The Stanford article on inconsistent mathematics goes into some of the motivations.

Pure mathematics has no empirical content so it cannot be a science.

Nothing in a purely mathematical formalism corresponds to a physical entity or process.

Bob Kolker

bobkolker wrote:Pure mathematics has no empirical content so it cannot be a science.

It cannot be an empirical science, but it can be a rational science.

bobkolker:
If that's the definitional requirement of being a science, then that's it. It's the monster tool of science, then. And producing a lot of it only costs what it costs to put things into people's brains.

I do believe that what most English-speaking mathematicians call 'pseudomathematics' can be a useful science and I'm reasonably certain it is but is only so in private and semi-private channels so far. I don't have time to continue today, and it's off-topic anyway.

Teuton:
Okay, continue.

Zigmen wrote:
Anyway. Is math a science? Or is it a tool?

I would describe it as a language. It deals with the intangible and abstract, so cannot be classed as a science in the same way as Physics, Chemistry and Biology. All the symbols of Math are two dimensional representations of reality. They themselves don't actually exist. For example, you could not show me 67. You could show me 67 of something, such as 67 marbles, but you couldn't show me just 67 om it's own.

It has language as a component, but I wouldn't agree that it is a language (anymore).