What is the formal definition of mathematics?

Question

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?

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 08 '18 at 06:08
[Here is the help page](https://philosophy.stackexchange.com/help). — Not_Here, May 08 '18 at 06:09
Math (as well as : science, philosophy, art,...) is very difficult to be defined. At least, we have to try to separate the human activity (the disciplines ?), the product of that activity (theories ?), the "pieces of world" (if any) that the disciplines are studying and that the theories are describing. — Mauro ALLEGRANZA, May 08 '18 at 06:53
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 08 '18 at 07:40
The core of mathematics is certainly correctly described as "an exact abstract analogy of reality". But there are extensions in several directions. Zeno has tried, inappropriately, to apply infinite series in linear problems and already Augustinus and Thomas Aquinas have pondered about God's mathematics and therefore left the secure domain of reality. In the last century the relation to reality has been cut by accepting counterfactual axioms. — Hilbert7, May 08 '18 at 08:55
@mb123: Did I criticize? It was my intention to agree, at least for that part of mathematics that in my opinion is the most important one and can be called science contrary to the other parts that are pure games. — Hilbert7, May 08 '18 at 17:48
I like Bertrand Russell's characterization. "Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true." A lot of insightful philosophy packed into that quote. — user4894, May 08 '18 at 22:47
@mb123 Given that the original question has had two edits, the context has been lost. Perhaps it is time to fold all the content into a single entry. — Mark Andrews, May 08 '18 at 23:38
@mb123 Much better. I cannot offer an answer, but others are going to have a better idea of what you are getting at. — Mark Andrews, May 10 '18 at 01:59
I find your elaboration quite acceptable +1. But perhaps you could refer to the fact that, although math *is* not reality, it can only be expressed and dealt with by means of tools of reality. — Hilbert7, May 10 '18 at 09:52
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 02 '18 at 23:19
I'm voting to close this question as off-topic because the question asks for opinions, which are opposed to the rules. — MichaelK, Jul 05 '18 at 08:22
@mb123 This section —the Q & A part of Philosophy SE — is not a forum. This is not a "Hey, I feel like chatting about these things, what are your **opinions** about this?" kind of place. That is what you use the [Philosophy SE **chat**](https://chat.stackexchange.com/?tab=site&host=philosophy.stackexchange.com) for. Have fun asking this question there. :-) — MichaelK, Jul 05 '18 at 10:10
Philosophy is not all about opinions, that's a very naive view of philosophy that a lot of people who haven't been exposed to real philosophy have. At the very least, if someone is going to argue for an anti realist position in philosophy where they think that none of the questions have any answers and it's all just opinions, at the very least, they'd still agree that the act of doing philosophy is not just sharing opinions. As Michael said, this is a Q&A site and it's part of the rules that questions need to be as objectively answerable as possible, we don't solicit opinions. — Not_Here, Jul 05 '18 at 16:01
Here is the direct wording of the close reason "Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise." You can see how your question falls into the ["Don't ask these types of questions"](https://philosophy.stackexchange.com/help/dont-ask) on the Help page of the website, again because it starts off being a very rambling "this is what I think" and then ends with "what do you guys think?" — Not_Here, Jul 05 '18 at 16:03
@mb123 Done! [Here it is](https://chat.stackexchange.com/rooms/79841/definition-of-mathematics). — MichaelK, Jul 06 '18 at 13:31
Possible duplicate of [What makes something mathematics?](https://philosophy.stackexchange.com/questions/25141/what-makes-something-mathematics) — Conifold, Jul 10 '18 at 23:15
I generally see math as a set of comprehensive languages stemming directly from the smallest possible set of terms. — Misha R, Jul 10 '18 at 23:50
@mb123 No, I don't think mathematics stems from reality. I think that *the history* of mathematics stems from reality (i.e., practicality), but mathematical systems a different matter. The foundational axioms of a mathematical system are always technically subject to being simplified, proven inconsistent, or otherwise updated. If that happens, the entire system is subject to change, or a new system can be born. And this can be done abstractly, without reality in any way. Mathematics may have started with practical reality historically, but has since become solidly grounded in the abstract. — Misha R, Jul 12 '18 at 02:58
@mb123 ...That said, mathematics being historically grounded in practicality may be the reason why math tends to do so well in describing the way the universe works. But, since we do not have another universe to test this on, there isn't a way to say whether or not our math could adapt to a wholly different structure of reality. I think the best thing to do here is to say that the term "stems" has two meanings - historical and systemic - and that the answer to this question depends on context. — Misha R, Jul 12 '18 at 03:12
@mb123 You lose me when you say "moving down the TOA back to reality." Once established, the roots of the TOA are separate from reality - and, if possible, can and will be abstracted and simplified further without the necessity of checking if the simplification is practically applicable in any way. The roots if the TOA are historically tied to reality, but systemically are not. — Misha R, Jul 12 '18 at 16:27

RodolfoAP · Answer 1 · 2018-07-05T04:10:55.363

1

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.

edited Jul 05 '18 at 04:10

answered Jul 05 '18 at 04:03

RodolfoAP

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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. – user33180 Jul 06 '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. – user33180 Jul 06 '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. – user33180 Jul 06 '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 07 '18 at 09:59
@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 07 '18 at 16:04
I think that I also have to include "tool" part in my definition. @Steven Could you please give me an example of a part of mathematics which my definition or his description does not include? – user33180 Jul 08 '18 at 17:55
Okay, so here the longer version: The definition of "Systems Theory" can be found here: https://en.wikipedia.org/wiki/Systems_theory. I don't see how this is in any case related to what you are writing; maybe you use a different definition? On 2.: I took your approach is that mathematics is a tool to describe the world. I don't think that this is what mathematicians do; there are many fields of mathematics not arising from physics. At least this seems to me as a controversial statement that needs more clarification. – Steven Jul 10 '18 at 20:51
On 3.: You say that commutativity is causality. I pointed out that one can do mathematics on non-commutative rings. I didn't make a statement on quantum mechanics, I asked a question. – Steven Jul 10 '18 at 20:54
@Steven: 1-your article: "System=organized entity made up of interrelated and interdependent parts". Ergo, an apple is a system. We do math with things (=systems); need a formal definition of an apple? cf. systems theory. 2: "There are fields...not arising from physics"...obvious! but math is based on physical experience; 3. commutativity is causality, but causality is not always commutative: you cannot feed a cow with milk and expect her to generate pasture; cannot expect all math objects to be commutative; can do causal processes with non-commutative objects; I don't see the problem. – RodolfoAP Jul 13 '18 at 03:06
@Steven: QM is relevant in the math philosophical concept due to it breaks our notion of causality (we've learned the basis of math from the causal nature): how is it possible for an event in the future to change the past? (Wheeler, 1978); some systems can travel faster than light. Causality is a sequence in time (past action->future reaction), but not in that direction (future action->past reaction)! Our understanding is broken! Worst even... math predicted such behavior! – RodolfoAP Jul 13 '18 at 03:17
I‘d prefer not to have a discussion in the comments, so I‘ll just say this: As far as I see, this is your own personal theory that you present in your book. I think there might be some issues with it, e.g. the broadening of technical terms (as commutativity, causality etc.). But it is your theory, and I think, you should specify this, as you would do when explaining the ideas of Schopenhauer for instance. This helps the reader to put it in context, since your theory is not consensus. – Steven Jul 13 '18 at 09:47

score 0 · Answer 2 · answered May 08 '18 at 18:24

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. . . — , May 09 '18 at 11:01

What is the formal definition of mathematics?

2 Answers2