Can the entirety of mathematical pursuits be logically reduced to a = a? All those subject areas, all those books, all of it, just one simple tautology.
If it can't, why? Would you say maths is informative? Does it tell us something about the world? Does it assert anything, and if so, what?

Throne777

Mathematical stuff like the Fibonacci Sequence, the number pi, and the golden ratio (phi) are found in nature. So yes, math can tell us quite a bit about the world.
One view is that, the universe is so perfectly put together that you can actually put so much of it into a mathematical equation. How could that be random?

magic numbers

Perhaps mathematical analysis could, but that's not all there is to maths. What about linear algebra? Geometry? Game theory? And so on?

Those are very far from a == a imho.

Which of them cannot be explained mathematically?

Actually a lot of the whole 'Phi is the number which the universe conforms to' thing is nonsense. The whole thing Dan Brown said about how you could divide up body parts and get to that number doesn't work.
What do the numbers tell us about the world though? What does the Fibonaccia Sequence tell us about the world? What does it assert?



Is geometry about anything? A.J. Ayer argued (and I agree with him) that geometry is that the 'axioms of a geometry are simply definitions, and that the theorems of a geometry are simply the logical consequences of these definitions. A geometry itself is not in itself about physical space; it in itself cannot be said to be "about" anything. But we can use geometry to reason about physical space' (A.J. Ayer - Language, Truth & Logic).

Do you agree? If not, why?

Throne777

Maths is a tool used by other scientists in their production of ever more details descriptions of the world.

I know that some people see maths as a valid pursuit in itself. I think it is like chess - some people thing its a fun way of spending the time and some even make a career out of it but it does not advance our understanding of the world. Mathematicians useful work in that they are finding new tools.

It is usually informative. Being able to express a phenomenon in a mathematically complete way (as opposed to only limted aspects) seems to demonstrate mastery of understanding. A lot of the trick seems to be being able to quantify things in the first place. Certain things seem unquantifiable [quantify love or pain in a complete way as it is felt?]. Many things also seem impossible to fully express through mathematics alone. Think of a dice with 1,000,000 (or even unlimited) faces. Mathematically expressing the "thing' in question in the way we usually do being only one face. Although a more complete representation might be possible, it elludes us currently or is not focused on.

It asserts a quantifiable universe if it is projected to be valid limitlessly. It is also highly dependent on application. The way it is applied varying greatly depending on the "senses" of those using it. The choice of certain distinctions based on this variable or that meaning a lot, although internally the rules may be the same....or not.

42- nuf said

I find these to be distinct questions, although you seem to think that they are all synonymous. Being informative and telling us something about the world is a very different thing that independently asserting something. Many things are informative; many things tell us something about the world by making intelligible something that would otherwise be unintelligible.

This does not mean that those things are asserting something independent of outside facts.

I think math is one of these things. While pure mathematics may not create new propositions, it renders the propositions of other disciplines intelligible. And you can only reason about something if it is understandable; therefore, mathematics is useful and informative in that it faciliates that understanding.



This quote supports that interpretation, so naturally I agree with it. While Ayer is only talking about geometry, I think the sentiment can be applied to mathematics in general.

It's easiest to pick on geometry because geometry is the discipline that is simultaneously the most abstracted and the most concrete. Geometry describes physical objects in a way that is more immediate than other mathematics. Algebra, calculus both also describe physical phenomena, but at a one-step remove. Describing a two-dimensional line or the motion of an object is harder to grasp than describing a shape. (Or perhaps that's just the wacky way my mind works? I always found geometry to be the easiest discipline to grasp, anyway.) At the same time, geometry describes these physical shapes in highly abstract ways, using numbers and formulas and axioms and theorems. This is why people struggle with it. A triangle is a triangle, not (h*b)/2.

However, all mathematics has this problem.

Still, the ability to reason about physical space is a useful tool, and thus mathematics is useful. It is also informative in that using mathematical concepts to perform this reasoning reveals things that one might (would) otherwise miss about that physical space.


I think you've contradicted yourself. Yes, mathematics is useful as a tool. But how are new tools discovered except in the pursuit of pure mathematics? Pure mathematics is more than a diversion; it is a useful pursuit in and of itself in that it facilitates the discovery of new tools.

Where did you get the idea that I assumed them to be synonymous from?
Maths tells us nothing more than how we manipulate symbols and the logical consequences of our own designs, but we would already know the latter if we didn't have such limited brains. As Ayer pointed out, if we had the mental capacities to immediately see the logical consequences of numbers, maths would hold no interest for us.



Geometry was mentioned specifically because it was one of Kant's a priori synthetic statements, the one which you would probably be most likely to agree with him on (if you are misleaded enough to believe in such things).

Out of curiosity, have you read Language, Truth and Logic?

Throne777

This conversation was totally finished at post #3