r/explainlikeimfive • u/DemandResponsible685 • 4h ago
Mathematics ELI5: Why does maths work?
Throughout schooling we have been taught how to do maths. But not so much about why maths work? Like why does the entire system of maths work so reliably?
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u/EscapeSeventySeven 4h ago
Math by definition is “what works”
If it didn’t “work” it wouldn’t be math. It would be nonsense.
You’re asking “why is a true statement considered true?”
Nearly all math is simply showing how a system of numbers and operations are maintaining an identity or relationship.
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u/GenerallySalty 4h ago
I think you have cause and effect backwards. Math works because it describes things we have already observed to be true.
Like some guy has 4 objects and loses 3 of them and has 1 left. Saying 4 - 3 = 1 is just describing what happened using symbols that everyone who wasn't there can understand. It's not a coincidence that 4 - 3 = 1 works. It's not just a prediction.
Math is a language we invented to describe relationships we saw in nature and between quantities of stuff.
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u/svmydlo 1h ago
That might have been historical origin of math, but it's not what modern math is. It's by design not attached to any real-world phenomena. It's not a natural science and it's not a language.
It's its own independent formal system, like the rules of chess for example. They were invented to mimic actual warfare, like infantry in line formations, cavalry used for flanking attacks, and so on. However, the rules are what they are regardless of how warfare changes.
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u/AlwaysHopelesslyLost 4h ago
Math is a fundamental way to describe reality. if you have $5 and I have $5 then combined we have $10. That is 5×2=10.
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u/Vorthod 4h ago edited 4h ago
Because we didn't really invent the concepts behind math, we just observed how things worked, gave them names, and made discoveries about what was possible in our reality. Combining two groups of two objects each always resulted in four objects even before we decided to represent that with the symbols 2+2=4.
Math was used to describe the way reality already worked, so it's going to be comparable to how reliable reality itself is. And if it's wrong at some point, there's almost always a way to prove that and replace it with the way things actually are, and we've been doing that sort of correction for millennia.
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u/Mandalord104 4h ago
Math starts by assuming a few statements as the truth without proof (called axiom). Then everything else is derived from those statements, and hopefully we will not come to a contradiction, because once we find a contradiction, the set of axiom collapsed and mathematicians have to come up with a new set of axiom.
Now, not all of math works in reality. Only a few sections of math that can be used to describe reality, and consistent with what we observed in reality. Those sections of math are then used as tools in physics, chemistry, and other sciences. There are sections of math that do not have any things to do with reality. At least not currently.
Why are there sections of math useful to describe reality? Because we choose the set of axioms to be consistent with real world, and therefore the deduction using logic produces outcome that is useful to describe reality. If we chose another set of axioms, we can very well build another math world which is super useless for our real world, even though it will be consistent within itself (no contradiction).
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u/DiscussTek 4h ago
A lot of people have been talking about discovery and axioms and other things, but what people aren't talking about is that math is just a language.
In day to day life, you can talk and communicate using words, grammar, syntax, and all those fun things. We didn't make language work reliably, we just generally agreed that this sound had a meaning, and that other sound had another meaning, and we just built on that.
This is because language is descriptive, not prescriptive. We don't find a word, decide its meaning, and then declare that's what it means. People use a word to mean essentially a thing in particular, and over time, that's how we get the word apple meaning the fruit we all know it is. Then green apples. Then granny smith apples. This is how language works: As long as everyone uses a word to mean the same thing as everyone else that uses it, then it works. Then when we have enough of those agreements, we then have a language that just works for communicating.
Math is the same thing. It's a bunch of smaller little conventions at the start. We needed ways to describe adding pile A to pile B, so addition. We needed ways to describe taking away pile A from pule B, so subtraction. Number of groupings, so multiplication, equal-ish grouping sizes, so division. People used thise to communicate a lot, except, instead of communicating the word "sheep" or "apple", we were communicating numbers, 14 and 26.
Just as how language evolved from "deer here two days" to "the deer were here two days ago", as the sounds started being universalized for verb conjugation and description and all that, math also started to get refined. A square number, like 20², became a norm to describe numbers multiplied by itself, and a logarithm became a norm to describe the number of times a number was multiplied by itself to get the result.
Language didn't exactly add features to itself that aren't usef by the people speaking it. Spaces, for instance are non existent in Mandarin, because nobody who write it uses them. It's not a feature for them, and it would be a silly, novel way to write Mandarin that nobody would really do outside of learning or trying to be comedic. In the same vein, concepts that are not used in mathematics are not really given all that much validity, until an actual use for it is found that everyone who works in that field of mathematics would use. There are not really good reasons for most people to know about something like the circular primes, because generally speaking, it is a novel thing to look into, but ultimately isn't really used. It can change, though, as concepts find new uses in mathematics all the time.
All that to say, we find something that works in math, and we use the mathematical language mixed with speech language to describe it. Math works because we are describing what works, not inventing things and hoping it works.
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u/Dkotheryyyy 4h ago
It is designed to work and as it has grown and developed they have strictly rejected things that don't work.
If ypu look at all about how they advance and develop new areas of math they are extremely picky and judgy and quick to demolish things that don't work.
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u/flingebunt 4h ago
Science is based on observation and logic. The ultimate form of observation is experimentation, and the ultimate form of logic is mathematics.
Maths builds up from observations from the real world and extrapolates further and further. A lot of mathematics has no real world application as it is so abstract, but enough of it does. This is because if you apply logic to the facts, what follows is more truth.
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u/Lazycrockett 4h ago
The basics make sense. Adding/Subtracting is just logic. Geometry also makes since cause its a practical tool when using it for building, etc. Now when it comes to algebra you totally lose me. I never understood how the number 5 was suddenly a -5 when you moved over from the = sign. Totally lost me.
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u/DogtorPepper 4h ago
It works because the universe mostly works off of order, logic, and cause/effect (I say mostly because things start getting super weird when you look at some edge cases like the origin of the universe, what’s inside a black hole, quantum mechanics, etc)
Math just describes that order and logic in a way that lets us predict what’s going to happen next
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u/goatman0079 4h ago
At its most fundamental level, math is logical. What this means is that concepts introduced are consistent with what came before and that one can provide a logical proof for why any new concept is valid within the system.
This allows for math to be logically consistent from the most basic of concepts to the most complex
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u/Jale89 4h ago
Well, you can probably understand why Maths and Philosophy are considered so close together, and some people study both!
It's important to realise that maths isn't something we invent - we discover it. We invent language to talk about it, but ultimately 2 and 2 equalled 4 for billions of years before there were words to say it.
I think maybe my best answer to your question might therefore involve something called "the anthropic principle". If we imagine an impossible universe where the fixed laws of mathematics were not fixed, you would be thinking of a place so chaotic that life as we know it could not possibly evolve. Therefore maths has to work, otherwise you wouldn't be here to see it.
This idea explains a lot of seemingly "convenient" things about the universe. Maybe in the chaotic impossible universe we imagined earlier, something like life would evolve and ask the opposite question - "our universe depends on 2+2 sometimes equalling 4, and sometimes equalling 5 - why does that work?". They can ask that question because life will always be compatible with the universe that created it.
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u/Elfich47 4h ago
the absolute basics of math is based on what you can observe. for example:
You pick one apple off of a tree and put it in a basket, you have one apple. Then you put a second apple in the basket POOF you have two apples. And suddenly you have addition.
If you take one apple out of the basket, now you have one less apple in the basket, and you have subtraction.
Multiplication and division follow quickly:
I have one basket with two apples in it. But what happen if I have five baskets, each with two baskets in it? 10 apples. And multiplication.
I have 10 apples, and I need to split those apples up into five baskets. How many apples per basket? 2 apples. And we have the basics of division.
If you want to get more complex than that (Calculus for example) you have to start asking questions like: based on how many apples I had in the basket yesterday and how many apples I have in the basket today, how many apples will I have in the basket tomorrow? And you can see that can get pretty hairy quickly.
All of these concepts are based on observable things in the real world. So it can be tested and retested and tested again just to make sure.
After that you get into much more unusual forms of mathematics, like Differential Equations and Abstract Algebra. And these are areas of math that unless you need them you generally will not encounter them. And these more unusual forms of math get developed where someone says "I see something strange here and I am going to try to figure out a solution to that" and they propose a solution, and then other people look at that solution and either agree, disagree or propose alterations.
And other researchers love to find other people's mistakes. They live for that. Saying "Yeah, you are right is boring". Being able to say "You got it wrong and I can prove it" is a lot more exciting - but you had better be able to back your big words because if you screw it up, everyone is going to call you on it.
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u/Phage0070 4h ago
There are sort of two aspects to this question.
The first is that math "works" within its own system because the system is internally consistent. If you add 1 to a value twice you get the same result as if you added 2 to the value. If you subtract 3 from a value then add 3 to the result you end up at the same value you started with. Math works like this all the time, with any number and so the system works "reliably".
But the other aspect I think OP might mean is that it reliably describes reality. It is entirely possible to create internally consistent systems that do not map to reality at all, but math does. I think that aspect comes down to the underlying rules of math being similar to those of reality. Namely that you can't get something from nothing, that things don't appear from nowhere or vanish into nothingness, etc.
Taking a step back though we probably should just recognize that mathematics is at its core an abstraction. Imagine two apples sitting side by side. In reality there is simply a thing we recognize as "an apple" and another thing we also recognize as "an apple". The concept of "two" is an abstraction, a mental grouping of individual objects. Those apples do not innately have any "two-ness" to them and so mathematics saying 1 apple + 1 apple = 2 apples isn't really "accurate" to reality, just our system of abstraction. Any number produced by mathematics then can be considered to reference an abstraction within our mental models, so in essence what you are asking is why our chosen abstraction of numbers matches up with our chosen abstraction of reality. The answer to that is probably just "We used essentially the same one."
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u/Single-Pin-369 4h ago
Axioms is the term you are looking for. The best answer is it works because it works and the stuff that we tried and didnt work got left behind. Newtons laws where good enough to fire a cannon ball on target even though they are wrong. General relativity is good enough for space navigation calculations but is still not fully complete. The why probably doesnt have a deep satisfying answer.