A comprehensive introduction to relationship between math and experience

Question

I am a mathematician with interest in physics and pure logic and exists one problem: the connection between math and physics.

Math concerned on pure universal truths and physics concerned on inferential thruth so how can we could explain physics with math?

Is it even possible to reduce universal truths to inferential ones that conserve veracity on the context that we are valuating a proposition? Yes of course the better example is theoretical physics and we do not have to prove the existence of something real we have to ask why?

Could someone please guide me in these topics?

Answer 1

If we go back to the roots of mathematics — operations on natural sets (e.g. counting) and basic geometry — we can see that mathematics is based in the measurement of physical experience. Of course, the focus of study for mathematics quickly shifted to the more formal question of how we can systematically compare, relate, and transform measurements: thus the ancient Greek preoccupation with the relationships between linear and area measurements that created things like the Pythagorean theorem and the constant pi. As mathematics has progressed it has become more and more formal, obscuring that essential connection to measurement behind a wall of abstractions, but the principle still holds.

Physics infers certain principles of experience that we can assess through measurement; mathematics determines how we can work with those measurements effectively and consistently. The two are complementary, not identical.

Answer 2

This is a great philosophical question, as you are sorting out what truth really is!

This is a question whose answer depends on the metaphysics involved. For instance, what do you consider truth? There are general theories of truth (correspondent, coherent, pragmatic), and then mathematical theories of truth.

Model theory of truth is essentially a correspondent theory of truth.

In mathematics, model theory is the study of classes of mathematical structures (e.g. groups, fields, graphs, universes of set theory) from the perspective of mathematical logic. The objects of study are models of theories in a formal language. A set of sentences in a formal language is one of the components that form a theory. A model of a theory is a structure (e.g. an interpretation) that satisfies the sentences of that theory.

Formal languages are partnered with automata, and from this perspective, natural language and the brain roughly approximate to their formal versions. Math is essentially a language that computers use (in this case we are a type of computer). It's about manipulating symbols that have meaning.

Proof theory of truth is coherent which means to say that the meaning is much less important than the consistency of the symbols.

Proof theory is a major branch1 of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of the logical system. As such, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.

So, mathematics is a language that describes quantity, relations, operations, truth, directions, and shapes and the syntax and semantics of math give scientific truth claims with which to work.

Science, while being largely an inductive venture, does heavily rely on mathematics which is a deductive venture. Regarding your question:

Math concerned on pure universal truths and physics concerned on inferential thruth so how can we could explain physics with math?

Math and physics work hand in hand to provide certainly in the truth of claims precisely become some aspects of "reality" can be known deterministically, and others only probabilistically. Both math and science are ways of thinking that help us make choices and attain our goals, some of the most important being survival and reproduction.

The is a topic that is heavily visited in philosophies of science. See Wigner's seminal essay.

Answer 3

Physics is about what exists in reality and how it behaves. Maths is about abstractions. To the extent that a particular mathematical abstraction has the same properties as some aspect of reality, that mathematical abstraction can be used to help us understand physics.

Now, you write:

Math concerned on pure universal truths and physics concerned on inferential thruth so how can we could explain physics with math?

Physics is about universal truths. It is universally true that you can't accelerate a massive object from rest to a speed above the speed of light. Also, physics is not inferred because theories don't follow logically from data. Nobody has ever seen the inside of the sun but we still have theories about it. Our knowledge of physics is created by guessing about how the world works and then testing the guesses. Knowledge of maths is also created by guessing and criticising the guesses.

Now, you ask how maths can describe the world. The answer is that physical objects instantiate some mathematical abstractions. Any mathematical abstraction we can understand has to be instantiated in some physical object since otherwise we couldn't know anything about it. A proof a mathematician writes on a piece of paper relies on the physical properties of the pen, the paper and the ink. If the symbols changed every time he turned his back on the paper he would have to interpret such marks differently.

For more on this topic see "The Fabric of Reality" by David Deutsch, especially chapter 10, "The Beginning of Infinity" by the same author especially chapters 5,6 and 8 and "Proofs and Refutations" by Lakatos.

Answer 4

First of all there can be science without math.

But let's leave that for now and stay with the physics-math pair.

Nature of numbers : exact Vs inexact

In math:
The billionth or trillionth digit of pi is no less exact than 1,2,3,4 being 1,4,1,5

In physics:
Constants are commonly used with hardly 1 significant digit

g = 9.8m/s²

And I can't imagine more than 8 significant digits ever.

To say

Physics numbers are inexact and math numbers exact

is correct but grossly understates the gap. It's more like

• Physicist view : Math-numbers are only corner cases (zero-error) of physics numbers.
• Mathematician view : Physics numbers are not numbers at all; rather they are a fuzz around a math number.

Let us (informally) define a real number as dddd.ddddd.... Where the digits d before the '.' are finite and the digits after are infinite.

Now does 9.8m/s² mean literally 9.8000000...?? Obviously not. It's more like 9.8XXX...
Where the Xes are an infinity of unknown digits.

This is what I mean by saying

A physics number is a fuzz around a math number

Continuous

Physics at least from Newton to Planck implied a math model(s) that was continuous.

Math has no inherent preference for analytic (so-called) entities.

The key word being...

Model

I find the general level of the there are no i apples thread appalling.

Won't spend more on that now than to say there's a woeful lack of appreciating model as the link between physics (empirical data) and math (rational data)

• Physics and math are separate domains
• They are linked by models
• (sometimes called abstraction/representation)
• These models are
  • non trivial
  • inexact
  • falsifiable

Even the trivialest models eg counting, are non-trivial.

Take the case of counting apples

• Go from 3 apples to 3.5 apples
• And from apples to humans
• So are we ok with 3.5 humans?
• With the help of surgeons or executioners?

The merchant of Venice

The problem of an exact pound of flesh is well known as the center-piece of Shakespeare's Merchant of Venice. It's a non-triviality of even the number 1 that gives the heroine Portia her winning strategy:

Tarry a little, there is something else. This bond doth give thee here no jot of blood; The words expressly are "a pound of flesh."

"a pound" means exactly 1 pound! — analytically trivial, empirically impossible!

Number size

Compare important Physics (Chem) numbers:

• Planck's constant 10³³
• Avodadros number 10²³
• G 10¹¹
• Electron rest mass 10³¹

These numbers are laughably small compared to the googolplex, Graham's number etc

Brings me to the biggest and most fundamental distinction

Infinity

I think it was @mauroallegranza who gave a 1-word discriminator between physics and math (Or was it empirical and rational?) — infinity.

Math is always about the infinite physics always starts and ends in the finite

Answer 5

Math concerned on pure universal truths and physics concerned on inferential thruth so how can we could explain physics with math?

There is no intrinsic, fundamental difference between the theories produced in mathematics and the theories produced in physics.

There is in this respect no fundamental difference between mathematics, physics and what people say using spoken languages generally, such as English, Russian or Japanese. These are all formal expressions of some body of knowledge using some articulated language.

These bodies of knowledge are all understood as theories, whether in mathematics, in physics or in everyday life when we use language to articulate our conceptions of the world. Obviously, there are important differences, but these differences are not fundamental.

First, all languages are formal. You don't need anything to read English, for example, that you wouldn't also need to read mathematics. Essentially, this means that you can understand a text by reading it. This is in fact also true of a music score, or even body language, and not only the body language of humans but that of animals. And animals do it, too. Indeed, we "read" our environment in this way. We just look at it and we understand what is going on.

Most languages also use logic, and this is obviously true in Greek, Latin, English, French etc. In mathematics, physics and in everyday life using spoken langages, all theories are essentially logical constructs of the form "A implies B".

All languages are essentially used to make claims about the real world.

There are two important differences between mathematics and spoken languages, though.

First of all, obviously, modern physics relies on mathematics as a language to express theories formally, so what is true of mathematics will be true of physics.

So the first important difference is that mathematics is a symbolic language. This is crucial. The main point about using a symbolic language is to restrict the possible interpretations of any mathematical text to just one, something which isn't possible in ordinary languages, even for philosophical texts. I don't think I need to develop this point since it seems easily understood.

The second important difference between mathematics and ordinary languages is that mathematics is rigorous, and indeed extremely rigorous. This is made possible by the use of a symbolic notation. However, there is also a discipline required on the part of mathematicians themselves, with the 19th-century mathematician Weierstrass being one mathematician much more rigorous than others, and that therefore a symbolic notation is not enough in itself to ensure the rigour of mathematics.

Rigour to do what, though?

Ordinary languages were used initially only to communicate basic facts. However, the logical capacity of the brain progressively began to seep into spoken language and to manifest itself in the way that people expressed themselves. Most plausibly, a few isolated individuals, in different part of the world, must have started to use "arguments" as a means to convince others instead of using threats or cajoling. It worked and so people used it.

And this must have spread very fast to other individuals, most likely without anyone realising what they were doing. Apparently, only Aristotle noticed that there was a theory that needed to be articulated about human logic. Even before Aristotle, argument has become a major feature in mathematics, in science and in politics. One could say that modern politics, arguably born in Athens at the time, is the effort to make other people do what you want by using argument rather than brute force, or indeed war. The ubiquity of argument in Greece at the time made things much easier for Aristotle.

Thus, argument and therefore logic became, very early in human history, a key addition to the art and practice of most spoken languages. It should be said that logic became used in mathematics and science only because it was already a feature of ordinary languages. I don't even think early mathematicians thought they were using any specific type of mathematical reasoning. Rather, they used the same kind of argument, and therefore the same kind of logic, to prove mathematical results as they would use outside their mathematical work.

Thus, logic is a common feature to most languages. However, in the case of mathematics, rigour makes, obviously, a very important difference. However, the difference is not in the kind of logic, not initially at least. The difference is in the rigour, rigour made possible by the use of a symbolic notation and the discipline of mathematicians.

Thus, in both mathematics and physics, people limit themselves to statements that follow logically from accepted premises, something which doesn’t happen very often in ordinary life. And this is the consequence of being rigorous.

It should be also said that there is another key element to both mathematics and physics which makes a huge difference. Both mathematics and physics limit themselves to simple problems. This isn't specific to mathematics and physics either. Most professions should be seen as specialisations in problems that have been identified as sufficiently simple.

The reason for that is straightforward. Most people are unable to agree on any problem which is not simple and professions couldn't afford the luxury of the sort of endless discussions as seen in everyday life, and indeed in philosophy, metaphysics and religion. And politics.

There is of course a difference between mathematics and physics and it is also important. Physics is exactly the same as mathematics except for one crucial point. Physics limits itself to premises that are thought true because proven empirically, i.e. by observing nature. This explains why a mathematical theory may subsequently be successfully applied in physics.

However, the theoretical work is essentially the same, i.e. to produce logically valid theories. However, in the case of physics, these theories become ipso facto accepted as sound since their premises are accepted as true, at least until proven false by new observations.

The difference is in the fact that mathematical axioms are not necessarily thought true of the real world, whereas in physics they are. That is, they are thought true. At least until they are proven false. And, sometimes, mathematical axioms become thought true by scientists.

I guess this is a really important difference.

However, there is nothing done in mathematics or physics that is not done using spoken languages. The only real difference is the use of a symbolic notation, rigour and the simplicity of the problems considered.