What is the formal definition of mathematics? [closed]

REWRITE

I am curious if mathematics could be defined as: "exact abstract descriptions of reality".

I use "descriptions" in plural because there are multiple distinct mathematical views on problems which it is solving, even on the same one.
It is obviously abstract, and it is also exact, because it explains reality very precisely in comparison to other sciences. However, it can't be absolutely exact because of the uncertanty principle. We can apply the same mathematics in sciences that are exploring completely distinct occurrences in reality. A simpler way to understand this comparison between mathematics and other sciences: https://xkcd.com/435/

I dont't think that math is reality, because in that case it would have to be absolutely exact. I also don't think it is something "magical", but rather a tool that people came up with to describe reality on an abstract level. Math is also fun, artistic and beautiful and there are many extensions to it, which have this as their only purpose and aren't about reality. But if we look at why mathematics even exists - because people need descriptions of occurances in reality on an abstract level, so that we can apply them on different problems which we are facing. In addition, if we say that mathematics is science, the "reality" part of the definiton is correct because science is about "physical and natural world" (from definition).

I think that there is no agreement about the definition of mathematics because none of them describes mathematics on a level which is abstract enough. This definition might be better in that sense. It is probably not perfect, so please comment if you think there is something neccesary to add to the defenition. Maybe that it uses special mathematical vocabulary? Instead of "reality" the word "phenomena (observable events)" could be used. Also, "systems/abstractions and their dynamics" could be used instead of "reality". Maybe the best one would be:

Exact descriptions of abstract entites, their relations and their dynamics written in a formal vocabulary named "Language of mathematics", which are usually used as tools for finding the most efficient ways of manipulating systems in reality for achieving particular goals.

"system" here means: "abstraction which is agreed on by multiple people and measurement devices about its existence, therefore an abstraction which is objectively observable."

This is also compatible with quantum mechanics from which it seems like consciousness is a clasifier for existence. But just one consciousness can't define objective existence because of possible errors in perception like halucinations.

Almost any science could be defined in a similar way. So if this is the definition of "science", then mathematics could be: The most abstract and widely applicable science.

I would like to add that philosophical questions like "what is the definition of mathematics?" are meant to be answerd just like any other questions. Ancient philosophers didn't have access to internet for instant fact checking and didn't know about new scientific discoveries like uncertainty principle, so from their perspective those questions really seemed unanswerable.

EDIT

Another way I think about mathematics: Mathematics is like a window (or rather "windows") to reality. But we don't really know if reality and window is the same thing.

So it's like we are taking a bottom up approach to describe reality. But I don't think that things like awareness and consciousness can be explained by mathematics.

EDIT 2

I removed the parts which made this question seem like it is just about "am I right?".

The question is:

What is your opinion on that idea and/or which definiton of mathematics do you like the most and why?

EDIT 3

Just because I don't want this question to be closed because it asks for opinion, the question is:

What is the formal definition of mathematics?

closed as off-topic by Not_Here, MichaelK, Mr. Kennedy, virmaior, ConifoldJul 10 '18 at 23:15

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "Questions that push a personal philosophy with no question beyond "am I right" or "what do you think" are off-topic here as this is not a blog. It's ok to express unique opinions, but you must have an actual, answerable question to go with them." – Not_Here, Mr. Kennedy, virmaior
If this question can be reworded to fit the rules in the help center, please edit the question.

• Hello and welcome to Philosophy.se. You can see the types of questions that are encouraged and the questions we do not accept on the help page. Your question is not currently acceptable as it falls under the "Questions that push a personal philosophy with no question beyond 'am I right' or 'what do you think' are off-topic here as this is not a blog. It's ok to express unique opinions, but you must have an actual, answerable question to go with them." reason. The question you're asking is virtually just "do you guys think my idea is right?" which is not what this site is about. – Not_Here May 8 '18 at 6:08
• Math is not essentially about reality. I'd call math as art of theorem derivation. Why art? You can choose any axioms like musician can choose an instrument and then do math. – rus9384 May 8 '18 at 7:40
• This question is still just "am I right?" and therefore not on topic here. Your last line especially makes look like it's just rambling. This site isn't a blog, it's a question and answer site. – Not_Here Jul 2 '18 at 23:19
• Obligatory xkcd: Purity. – MichaelK Jul 5 '18 at 8:18
• I'm voting to close this question as off-topic because the question asks for opinions, which are opposed to the rules. – MichaelK Jul 5 '18 at 8:22

3 Answers

Mathematics - the logical and provable relationships of things

What I loved about computers, is you could define a world exactly not approximately and it did what you wanted, not maybe, but precisely.

And this is the power of Mathematics. In a real sense, things become provable as far as we can prove anything.

Maths has got flaky in some areas, but even here, what appears to be impossible, has actually be reflected in the real world, which has led some to suggest everything is just a mathematical model, rather than maths is copying the world, or approximating it. In this frame of reference, if you could summarise maths your would define existence.

• It's been done, if I get what you mean. GS Brown does it in 'Laws of Form' with his 'calculus.of indications', which is a model of how form arises from formlessness. Lao Tsu also gives a mathematical summary of origins. . . – PeterJ May 9 '18 at 11:01

Math is two things.

• A language, which allows us to describe our past perception in an objective way. When we perceive something, we can associate it with ideas that have a correspondence in mathematics. So we are able to count things (6 apples), name things (apples are x, oranges are y), describe groups (6x +3y), etc. etc. We can express heavily complex perceptions (e.g. the wave function) using math. So, it helps communicating. Remark that the word "past" was used.
• A tool, which can be difficult to master. But when done, allows us to model the future of things. What will happen (future) if you buy one apple and one orange from the group described before? Voilà. We've predicted the future.

Why the words past and future? Why the word thing?

Inherently, math depends on systems (c.f. Systems Theory). Things are essentially systems, or groups of parts. If you have an apple, it doesn't really exist in nature. There are no atomic boundaries between you and the Apple, if you grab it with your fingers (see cold fusion). Everything is just atoms. So, what is an apple? A system (a group of parts). The parts of an apple can possibly be its skin, a small branch perhaps, the flesh and some seeds. You might wonder, what is the skin? Well, all systems are just perceptions. That is what math is about. The mathematical language is based on systems.

Now, the tooling side is based on causality, which is action and reaction through systems. You might have the f(x)=3x system (math tools are also systems!), input a 9 and the system will output 27. That is causality. And causality is always related to past (action) and future (reaction). If you write x=f/3 you are not going backwards in time. You are adapting a system to behave differently regarding the standard time sequence (do you remember thinking in school that commutativity was a stupid rule? Well, it isn't! Without it, there would be no way of reverting a causal sequence, just as we did!).

Quantum mechanics generated new problems in causality. We've been able to describe the quantum physics theory with math but things (systems) proved to be completely different in reality, compared with what we perceive. Quantum physics experiments show that our perception can not only be wrong about the state of a thing (things can be in two states simultaneously, can dissappear and appear in a different place, can be mesured only in part... What???), but that we can even change the past. So, our perception can be wrong. The systems theory might be wrong. But math keeps allowing us into the real nature without errors.

Perhaps the most amazing thing about math is that it is not perfect by itself, but describes something absolutely perfect: nature.

• I agree, but I think that "describe our past perception in an objective way" and "allows us to model the future of things" is included in my definition. "describe" is included in "analogies" because analogy's purpose is "explanation" - description is a part of explanation. Modeling future is also a part of explanation - if you can explain occurances (statical and dynamical) of reality, you can also predict what happens after them. If we replaced "reality" with "phenomena" it would also emphasize that it is about our perception ("observable" events). Systems are just entities in reality. – shp999 Jul 6 '18 at 10:17
• There are also purely imaginary systems but those aren't really useful, they are just something to play with (until somebody finds it's application in reality). And systems can be split into subsystems or dealt with as a whole - we define those "boundaries" - it just depends on what level of abstraction are we on while tryng to describe reality. – shp999 Jul 6 '18 at 10:21
• "So, our perception can be wrong. The systems theory might be wrong. But math keeps allowing us into the real nature without errors." I agreee, this is why I mentioned that those analogies are not absolutely exact - they are wrong to some extent. – shp999 Jul 6 '18 at 10:34
• Several comments to your answer: 1. This is not what „Systems Theory“ means. 2. I think you confuse mathematical physics with mathematics as a whole. 3. What you described is absolutely not causality. One can do mathematics on non-commutative rings. And why is Quantum Mechanics relevant to your answer? – Steven Jul 7 '18 at 9:59
• @Steven 1, 2, 3 and QM: statements without arguments. Mine are on the text, read them. Please post an answer with the right approach so we can learn. Cheers. – RodolfoAP Jul 7 '18 at 16:04

Currently there is no definition of mathematics on which everybody agrees.

From wikipedia:

Mathematics has no generally accepted definition. Different schools of thought, particularly in philosophy, have put forth radically different definitions. All are controversial.