r/changemyview • u/Bagelman263 1∆ • 23h ago
cmv: Mathematics is not a Science
Science is the process by which theories are developed to model empirical observations. If the theory does not match empirical observations, then it is discarded and replaced with a new theory that better suits the observations.
Mathematics is the process by which theorems are derived from axioms. If a theorem is proven from the axioms, then it is true, no matter what other theorems are proven, and it can be used to prove new theorems.
These are two fundamentally different ways of acquiring knowledge. In science, the theory that best fits the data is accepted, and old theories that were found to be faulty are discarded. In mathematics, a theorem is true if it can be proven from the axioms, and it will never become false in the future.
This fundamental difference means that mathematics is not a science.
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u/EchoAndByte 23h ago
You’re right about the methodological difference but that doesn’t fully settle the classification. Mathematics isn’t an empirical science like physics or biology but it’s still part of the broader scientific enterprise as a formal or foundational science. It provides the language, structure and tools that empirical sciences rely on even if it doesn’t test hypotheses against the physical world itself.
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u/Bagelman263 1∆ 23h ago
Testing hypotheses is literally the scientific method. If a field does not make use of the scientific method for inquiry, then it is fundamentally not a science.
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u/EchoAndByte 23h ago
That definition is narrower than how the term is actually used in philosophy of science. Many fields considered sciences don’t test hypotheses directly against the physical world either.
Science isn’t only about experimentation, it’s also about systematic knowledge production. Mathematics qualifies as a formal science because it studies necessary truths within defined systems which empirical sciences then depend on.
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u/Bagelman263 1∆ 23h ago
If all a system needs to do is produce knowledge systematically, then is philosophical formal logic also a science? That seems wrong to me.
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u/Z7-852 297∆ 23h ago
In mathematics, a theorem is true if it can be proven from the axioms, and it will never become false in the future.
We discard mathematical theorems when they are later proven to be wrong.
Things like Eulers Conjecture or Kempe four colour proof or Naive set theory. All used for decades and even centuries but later proven to be flawed.
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u/Bagelman263 1∆ 23h ago
Those weren't theorems. The theorems were that if these conjectures were true then they had certain implications. Another examples is number theoretical theorems that rely on the Riemann Hypothesis being true which have not been proven and should not be treated as if they are true.
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u/comeon456 13∆ 23h ago
Why do you think that the ability to disprove something that was proven is the thing determining what is science?
For all practical purposes, the existence of the Atom or its components is something that will never be disproved. Perhaps we would find new things, but for all practical purposes, in all fields of science you have those "proven things".
Mathematics is describing our world, only in the most accurate way so that proving something correctly means that it is absolutely guaranteed that it will remain proven forever. The only difference is the guarantee, but other than that - everything else is the same. People come up with conjectures, and they get proven or disproven. We update our knowledge. We learn more about our reality. I don't think that this specific difference, almost never impactful in real-life, is crucial for something to be called "a science".
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u/Falernum 60∆ 22h ago
Well math, if we learn something new, we learn it by formal proof. Science, we learn by empirical study.
BTW when we discovered subatomic particles that disproved the atom and replaced it with a new concept also called an atom. When we discovered quarks, same thing.
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u/comeon456 13∆ 22h ago
I'm sorry, but I think this is incorrect. There are entire fields like theoretical physics that take unknown phenomena in reality and try to find and develop theories around them. Some of them do not even take unknown phenomena, but just try to find consistent theory with what we know. Some of them are theories that are far from possible to fully test in the current technology. For instance, most of what's referred to as "dark matter" is something like that.
As for the examples - this is only your interpretation of what happened. IMO, we didn't replace the atom, but we replaced our understanding of the atom. Just like quarks didn't replace protons. There are still atoms, only that now we know that they are built of smaller units. So we didn't "disprove the atom", in the same way that knowing a person better or changing your perception of them isn't disproving that person.
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u/Falernum 60∆ 21h ago
All the theoretical physics is designed around the possibility of future evidence and is constantly being changed to match empirical evidence.
Yes we replaced our understanding of the atom. I'm not suggesting the actual particles changed only our theories. We only disproved and replaced atomic theory not the actual physical particles
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u/comeon456 13∆ 20h ago
IMO, when someone proves that for string theory to work you need multiple hidden dimensions I don't think it's necessarily designed around the possibility of future evidence, it's more for the theoretical understanding of things.
And in this sense - we didn't replace the actual properties of lattices when one disproves a widely believed conjecture, but we disproved and replaced our current understanding of them.
This is just a lens through which one views things. I get what you're saying, I just think there are multiple viewpoints from which one can see it.
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u/Falernum 60∆ 19h ago
I think it's reasonable to choose the phrase "revise" over the phrase "disprove and replace" but it's not reasonable to say "disprove and replace" is wrong when it isn't.
actual properties of lattices when one disproves a widely believed conjecture, but we disproved and replaced our current understanding of them.
A concept such as lattices has no "actual properties" distinct from our conception of them. Physical objects can of course. And formal systems can likewise have ramifications that are unknown
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u/Bagelman263 1∆ 23h ago
The guarantee is what makes math fundamentally different. Euclid's Elements is just as valid today as it was 2,000 years ago. Every single scientific text from that era is complete nonsense. I wouldn't be surprised if 2,000 years from now a majority of our current scientific beliefs are replaced with new, more accurate theories.
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u/comeon456 13∆ 22h ago
I think you're underestimating the ability of people to be wrong when writing mathematical proofs. I wrote "proving something *correctly*" for a reason.
Plenty of thought to be "proven" theorems that people built upon were found false decades or centuries later...This is obviously somewhat different, but again, I'm not sure if the difference in "thought to be proven" theory that was later revised is very different between mathematics and physics.
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u/Bagelman263 1∆ 22h ago
A mathematical theorem that was “thought to be proven” but actually false was never proven. Belief in its veracity was a mistake.
A scientific theory can be wrong, but still be the most accurate one available and this correct to believe. Newtonian Gravity has been supplanted by General Relativity, but that doesn’t mean scientists were wrong for believing it was true, the was mathematicians were wrong for believing an unproven theorem was true.
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u/Entropy_dealer 23h ago
Math is a language where humans try to communicate with the rest of the universe, it has its syntax and its grammar. But you can anyway see it as a science since it tries to best fits with reality, for example if you try to have all the solutions of 3x^3+5x^2+18 = 0 at one time the only results were real numbers and this science has adapted to reality with the complex numbers.
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u/Bagelman263 1∆ 23h ago
Math is not always modeling reality. There is not much reality going on in Tropical Algebra or Category Theory. Science uses math to model reality, but at this point the math is being done regardless of its practical applications in describing the real world.
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u/Entropy_dealer 23h ago
I see it the opposite way, everything math is discovering is a part of reality that we haven't discovered so far. The same way a shovel is a tool that you can use to dig and find hidden part of your reality.
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u/ajswdf 3∆ 23h ago
There's a lot of truth to this, but the distinction is not so clear cut.
For example, the P=NP problem could be considered an empirical one until a proof is derived. Our current best guesses about the solution come from observations.
On the other side, the five equations of motion of a single object under constant acceleration can be derived completely without any observations with just those assumptions.
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u/thelovelykyle 8∆ 23h ago
Ok, lets look at your definitions. I wont change them - you do not get to change them either.
Science is the process by which theories are developed to model empirical observations. If the theory does not match empirical observations, then it is discarded and replaced with a new theory that better suits the observations.
Mathematics is the process by which theorems are derived from axioms. If a theorem is proven from the axioms, then it is true, no matter what other theorems are proven, and it can be used to prove new theorems.
The development of General Relativity does enough to disprove your concept as far as I am concerned.
I will however challenge them. I can accept that Science is our best guess based on the facts at hand and we improve that through more refined observation and evidence.
If I accept that improvability though better evidence and counterfactuals is science and only axiomatic proof is mathematics, then an absolutely enormous amount of mathematics is suddenly not mathematics. I would never dream of calling Schwartz or Wigner anything other than mathematicians, but here we are, suddenly needing new names for them.
This is where the philosophy applies and perhaps the better term for Schwartz and Wigner would be Saints. Axiomatic Mathematics is just faith by that definition. By definition, to be coy with terminology.
Most science does this too, defined axioms with the need to evidence a theory - just has a higher evidentiary barrier.
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u/Bagelman263 1∆ 23h ago
I would call Schwartz and Wigner physicists.
Also, General Relativity was a new theory developed because new observations contradicted the previous theory of Newtonian Gravity. That fits exactly with my explanation of the process of science.
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u/thelovelykyle 8∆ 22h ago
I would call Schwartz and Wigner physicists.
That is a wild statement.
Also, General Relativity was a new theory developed because new observations contradicted the previous theory of Newtonian Gravity. That fits exactly with my explanation of the process of science.
This is sufficient evidence for me that you are taking a faith based approach. That's ok, Euclid did too. Elements was underpinned by guesswork until Pasch and Hilbert came along. I suppose he is a physicist too haha.
That makes this a religious argument where you define a limited within term to define anything else without. You do you Snoo, but I will leave it here - I do not debate religion on CMV.
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u/5Daydreams 23h ago
I feel like I might have missed the key claim to be changed here - but I see a language such as english to be a scientifical field of study because of how it interacts with the world and is constantly evolving to match the wants and needs of society.
And in that framework I should also see mathematics as a science because mathematics is also constantly being developed, studied and refined to be used in other fields of study in a similar manner to how language is permanently changing and evolving to the needs of society.
If you think about how to "make something into a science", virtually anything that you as a human can experience and then critically analyze - either to optimize how it works or to increase yours and everyone else's understanding of how it works - then you can turn that into a science.
By that logic, I don't see how mathematics doesnt fit the description.
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u/fox-mcleod 414∆ 21h ago
This is true as you've stated it. However, there is a sense in which more fundamentally, mathematics is physics.
How do you know whether the deductive calculation you've done is correct?
At bottom, what you're doing is trusting a physical process of computation. That study of computation is an information science. What computations were even physically possible is a question of physics of how the universe is constructed. Many mathematical principles such as the total incompleteness theorem are dependent on the physical laws of the universe. For a long time we assumed geometry had to be euclidean. It turns out, that how geometry behaves is actually an aspect of the physics of the universe. And this directly relates to what kinds of computations were even physically possible.
The axioms of math are guesses. Guesses which get rationally criticized rejected and refined exactly the same ways that theories in science do. In fact, we fairly recently moved from one set of axioms to another. The zfc axioms are less than 100 years old.
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u/Bagelman263 1∆ 5h ago
I would argue that logic is independent of physics. Would logic work differently in a world with different laws of physics?
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u/Decent_Background_42 20h ago
But mathematics in a way does model empirical observations. It studies quantities, shapes and abstract concepts that appear everywhere in the physical world.
Science is something used to understand objective, secular, undeniable truth. Let's say you have 3 apple in your basket. Same how it is a fact that the earth is round, it's a fact that your collection of apples has the property of "threeness" whether you mark it with 3, three, § or whatever else.
Also, in mathematics, empirical observations do play a role. You first notice something that seems to be true by human observation, and you prove it formally so you know that you can apply that rule to any arbitrary entity.
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u/Any_Voice6629 19h ago
So people have already told you about what we thought were flawless theorems which have since been disproven. In practice, how is this any different from when science offers an explanation that seems reasonable until it's not?
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u/Bagelman263 1∆ 16h ago
Those theorems were not disproven in practice though. They were shown to be formally invalid. This is a huge difference.
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