Can mathematics only be defined by indirect observation?

[Imparcticle]

When we study something, say math, we learn accepted axiomatic ideas. aren't those very axiomatic ideas what define math? By defining, do they not tell the precise nature of mathematics? Supposing the answer is "yes" to both questions, then allow me to ask one more question: Is "the nature of" something what that something really is? Can something only be defined by indirect observation? What really is mathematics? Could it be that, like in QM, we can only be definitely sure of one thing while not the other (and unlike in QM, it does not need to be restricted to particles in this case)?

 

[laserblue]

I question the overall line of reasoning here. I also question the first sentence. Let me rephrase it using "parallel lines" rather than "math". When we study something, say parallel lines, we learn accepted axiomatic ideas. Aren't those very ideas what define parallel lines? This sounds strange to me because axioms developed for parallel lines have proven inadequate when one generalizes beyond flat "Euclidean" space. So I would reply no to the first question. When we study something, say physics, we learn accepted axiomatic ideas (E.g. Newtonian Mechanics). Aren't those ideas what define Physics? See, here I could reply yes Newtonian Mechanics does define Physics (say up until 1911) and then No, Newtonian Mechanics does not define Physics because it is insufficient or inadequate to do so (circa 1900). The axiomatic ideas define someone's conception or model of the object of study not the object itself. It's all made more confusing by using the word "study" here as well. Axioms have often been the end result of an inductive process and a way of succinctly describing an adequate model in a way that can be uncompressed using deduction. One is studying then, someone's model, not the object. The model may be inaccurate. When we study something (in a textbook), say the movement of the planets, we learn accepted axiomatic ideas (Ptolemy's epicycles). Aren't those very axiomatic ideas (Ptolemy's epicycles) what define the movement of the planets? Now, Einstein did claim that 'there is no inductive method which could lead to the fundamental ideas of physics", but there is also no guarantee that the axioms of the proposed model correspond to reality.

 

[vanesch]

The axiomatic ideas define someone's conception or model of the object of study not the object itself.

This is a very important remark! It shows the borderline between formal thinking (logic, math...) and natural sciences (physics...). Physical concepts remain necessarily vague in their description, and only correspond to formally definined concepts within the framework of a model (which can be quite accurate or not). As I wrote a few weeks ago, the physical electron is very difficult to define (although most physicists "know" what you mean with an electron). On the other hand, the "electron" of the standard model can be formally defined (as a quantized Dirac field). We interpret this formally defined electron as describing the physical electron. In order for this to have any meaning, we cannot rigorously define what is a physical electron, because then we could, by pure mathematical reasoning, show that it is or is not, equivalent to the Dirac electron ; no experiment would be the judge.
So physical sciences NEED vaguely defined concepts to talk about "the real things out there" or they would stop to be experimental sciences and become a formal system.

cheers,
Patrick.

 

[wuliheron]

Words/concepts only have demonstrable meaning according to their function in a given context, and that includes the idea of "studying" and "math". What was not considered mathematics a hundred years ago, is today. Is mathematics merely whatever we define it to be, or is it some sort of "thing" independent of us? To take this idea further, is there such a thing as reality, or is it just your imagination?

 

[vanesch]

What was not considered mathematics a hundred years ago, is today.

What do you mean ?

cheers,
Patrick.

 

[wuliheron]

Fractal Geometry is one example. It is a sharp deviation from Euclidean Geometry, and a hundred years ago the very idea of a "fractional" dimension would have been considered utter nonsense, a complete contradiction in terms that has nothing to do with mathematics.

 

[laserblue]

I actually saw a paper in a journal along the lines of the original post here but I can't recall what journal it was. I'm thinking it was FOUNDATIONS OF PHYSICS.

 

[selfAdjoint]

Fractal Geometry is one example. It is a sharp deviation from Euclidean Geometry, and a hundred years ago the very idea of a "fractional" dimension would have been considered utter nonsense, a complete contradiction in terms that has nothing to do with mathematics.

Actually there was almost no resistance to the idea of fractal dimension when it was discovered (it wasn't called by the trendy name fractal then! It was Hausdorff dimension). There are always new discoveries in mathematics; that is what mathematicians do.

I don't want to give the impression there has never been controversy in math. Cantor's theory of transfinite cardinals was bitterly resisted, and the general abstractness of twentieth century math (the Boubaki program) was offensive to many.