Acknowledgments

On all the wonderful answers

Greetings to all of you

Yes, if we wrote the axioms of logical mathematics in right direction.

Hi Hassan: By "absolutely certain" do you mean that all mathematical questions are "decidable"? Even when the physics is simple (such as the Spectral Gap question in quantum mechanics) the mathematical results appears to be undecidable (Undecidability of the spectral gap, T. Cubitt et al, Nature volume 528, pages 207–211 (10 December 2015)).

The formalist tradition has been questioned, this consists in writting a system of axioms.

Gôdel proved the existence of undecidability of some statements based on axiomatic systems with the same complexity as arithmetic. Then the only way to decide them is to go out of the axiomatic system.

@Hassan Kamil Ressn

Physics is vast and Mathematics is also vast. However when we correlate

between them ,we find oh the nature is simple .What happens we are not ?

There are many math models where we can not correlate with physics and vice versa. I feel we are trying measure infinity with a scale of 30cms. Similarly if want to take weight of sea water in kilogram . Do you know how many Years it will take ? Your question can be related to above .Sorry

Wittgenstein dictum

Math is only a vast tautology whoes only meaning is a=a

That is what happens when it is without interpretation.

I think that each scholar sees the phenomenon of interest through his/her own angle (lense). Thus, mathematics could be relative in terms of a research context, I think. That is to say, a mathematical algorithm is absolute in its logic and scientific rigor. However, the results derived from the mathematical algorithm could be changed by a context. The mathematical algorithm itself could even be changed. Scholars should be mindful of their belief that their outcomes can be perfect.

I prefer the observations of Claire Su-Yeon Park and @Peter. They are both applicable. There is also social aspect of mathematics with regard to satisfaction of mathematical proofs in practice. Mathematics is not all settled and mathematicians do not escape human being.

For applied mathematics, there is an intended interpretation, and if it is about real-world affairs, the interpretation is into a contingent reality (and of course, is itself contingent with regard to internal-to-mathematics acceptance), and that matters. The math is not enough in those cases, and interpretations can be misapplied (e.g., in computer programming where the finite bounds on data are over-looked and also algorithms that have socially-unacceptable interpretations, a current topic of concern).

Acknowledgments

On all the wonderful answers

Greetings to all of you

The concept is an abstract structure, universal, inespacial and independent of the subject. It is always related to other concepts, always refers to objects whether real or ideal, the concept is the fundamental logical element: With it are built the judgments and with the judgments the reasonings. It is the principle of all knowledge and the first truth we have of things.

Mathematics is not at all a collection of logic rules. That can be only in certain of its branches as the Cantor transfinite numbers or so on. Mathematics has real objects as the real number pi or Pitagorian distance. Physics without Mathematics would be only a collection of recipes and it would be interesting to observe that Riemann Geometry was previous to the General Relativity and Differential Calculus made close to Newtonian Mechanics, and so on.

Tha mathematical concepts are absolute and the background necessary for formalating any scientific theory that we have because it allows to joint "reality" with "calculation" or "prediction" given a certain finite number of dates.

How mathematicians tangle themselvs up;

Start from a recognized but not so trivial identity, say

xx-yy = (x+y)(x-y)

or

xxx+yyy = (x+y)(xx+yy-xy) a favorite for fermat players.

Everyone knows how to simplify such identities, say set y=0 or 1 in any of these two.

You still have a valid identity.

What is not so well known is that they can be complicated (or generalized) to any desired degree one wishes

Now take the total differential of both sides of any such identity. Say of the first.

2 x dx - 2 y dy = (dx + dy) (x-y) + (x+y) (dx -dy)

Now just replace dx by a new variable u and dy by the variable v

Now there is a new identity involving four variables (x, y , u, v)

In a second round you do the same to get an 8 variable identity, and so on!

you can also replace u by any function of x and y, and v also by a different function of x and y,

it is still an identity.!

A hopeless game!

So any single formula you have, you never know the real number of real actors that you can have.

You usually simplify to the most simple answer, but you never really know.

Dear Peter,

I don't agree with many points that you have been telling about mathematics and physics. Mathematics is not only theorems, corollaries, demonstrations....what is fantastic is that these objects are closely related with reality. The concept of distance is a pure geometrical mathematical concept that is fundamental in physics and we measure the distances. This is real and all the concepts of geometry are very close to real concepts as curvature. General and Special Relativities employed abstract concepts as Riemann geometry. Gauss and Euler were two mathematicians who always solved real problems besides to proof interesting abstract theorems. Mathematics is an absolute branch of knowledge with is very close to the real world in spite to use very abstract tools, but not too more abstract as energy, electric charge, magnetic field, ....etc both are products of the same source: the human mind for understanding our world in a logic form.

Basically I agree that math is not necesarily logic, nor has to do with reality necesarily.

However, historically math ows a debt to both logic and reality, since you can start with applied math and later on turn it into a generalized abstract construction; and there are researchers in the field mathematical logic. Logicism is just one point of view in math.

It is just that abstract math without interpretation is valid, but perhaps does not help society that much.

Dear Juan,

Mathematics needs to follow necessarily logic rules and what is less clear is that follows reality. The problem is what we understand for reality and how we define it. There is not nothing more real that one equation which allow to know how certain phenomenon appears under a change of some given variables. It is more real that there are only five regular polyhedra than the recient change of year that everybody has welcome somedays ago!

It has rules, for sure, but these rules are not always logical rules, not derivable from logic alone.

You perhaps start from axioms, and all the rest is deduction, but you never get to prove the axioms.

So at least some element in it is just art, not logic.

Then you have the Godel problems with any formal system.

This is not popular, but impossible to totally disregard, you have to live with the limitations of logic itself.

Math makes reality appear more real than it really is? Could be.

Juan Weisz Perhaps another way to look at it is that reality has mathematics appear more real than it is. I still favor Einstein's view at a time when mathematical logic and axiomatics was rather young. My appraisal is aligned. http://orcmid.com/blog/2010/02/abstraction-einstein-on-theoryreality.asp

Poincaré was one of greatest mathematicians without and doubt and he believed that the structure of non-Euclidean space can be known analytically. Poincaré held that convention plays an important role in physics. His view (and some later, more extreme versions of it) came to be known as "conventionalism". Poincaré believed that Newton's first law was not empirical but is a conventional framework assumption for mechanics. He also believed that the geometry of physical space is conventional. He considered examples in which either the geometry of the physical fields or gradients of temperature can be changed, either describing a space as non-Euclidean measured by rigid rulers, or as a Euclidean space where the rulers are expanded or shrunk by a variable heat distribution. However, Poincaré thought that we were so accustomed to Euclidean geometry that we would prefer to change the physical laws to save Euclidean geometry rather than shift to a non-Euclidean physical geometry.

I guess I could only say about this that if a physicist talks about a noneucledian space, he means some sort of curvature

that is both in the math and in reality, but in distinct ways.

Just like when a car goes on a curve, you are pulled to the outer side of the curve, the ocupants of the space ship would feel a force pullig sideways, so this is real and can be detected.

Right now cannot recall exactly, but one would say for example that the Ricci tensor is nonzero, on the math side.

There is no real confusion, if they do not coincide, something is wrong, but the idea is in both.

The study of real numbers may have changed, but the essential intuitive ideas of space and the essentials of a smooth curve

have not.

The best, juan

Are you confusing the mathematical word "space" with the "Space" of the real world (TM).

Is TM the tangent manifold? Is this the real Space? For me a real space can be perfectly a Principal Fiber Bundle as a vectorial space or cotangent space T*M. What is more real in the usual 3-D Euclidean space more than these other spaces? What is the definition of real for you?

Well this has become tricky. SR means that the length of a rod depends on the speed it moves past you.

Its not just the space but the dynamics also.

Depends on the whole initial conception, or theory, that the mathematician might use. (Cannot hold them entirely

responsible for this) The math people do not have real space, would call almost any of their constructions a space.

Mathematical entities are by their nature abstractions. Absolute and relative do not seem to be apt distinctions.

With regard to mathematical formulations that are taken as theories applicable to nature, that is an empirical determination. Newtonian physics and mechanics works for a certain level of the observable world. That it was found inadequate at levels where relativity applies is not alarming. How could that have been known in Newton's time? Just how science develops.

Einstein himself was very clear about the difference between mathematical theories and application to natural situations. He gave a nice lecture, "Geometry and Experience," about it almost a century ago. I have references to that in a blog post, http://orcmid.com/blog/2010/02/abstraction-einstein-on-theoryreality.asp

(expanding on my previous comment)

If you say vector, people know what you are talking about.

But if you just say number, there is no technical definition.

There are too many definitions going around, for my taste.

With something like Hilbert space, you eventually get used to the idea by using it, but if you just read the definition, you get almost nothing, unless you have a lot of previous exposure.

Mathematicians talk about continuity or discretness, smooth curve as intuition, not reality.?

For us it is a matter of scale or observation. Thus the flow of a liquid seems continuous, the flow of sand grains as jumpy. If we examined the flow of water

at the scale of a molecule, it would also jumpy.

In math you would I guess use the real number system for a continium making up the shape of a curve. However, since the separation of the reals is hoplessly small, or even nil, for all practical purposes it does constitute a continium.

A perfect continium not imitable by physical means.

At least

from a scientific point of view, it would end the discussion.