The difference is that if something is proven mathematically it’s 100% certain and will not change. In other sciences you may be taught things that later turn out to be flat out wrong.
Bingo, I was taught in genetics class in the 1990s that RNA played a role but DNA was the primary driver and now my understanding is the current consensus is RNA is the primary driver.
Not quite. Science is empirical, which means it’s based on experiments and we can observe patterns and try to make sense of them. We can learn that a pattern or our understanding of it is wrong.
Math is inductive, which means that we have a starting point and we expand out from there using rules. It’s not experimental, and conclusions don’t change.
1+1 is always 2. What happens to math is that we uncover new ways of thinking about things that change the rules or underlying assumptions. 1+1 is 10 in base 2. Now we have a new, deeper truth about the relationship between bases and what “two” means.
Science is much more approximate. The geocentric model fit, and then new data made it not fit and the model changed. Same for heliocentrism, Galileos models, Keplers, and Newtons. They weren’t wrong, they were just discovered to not fit observed reality as well as something else.
A scientific discovery can shift our understanding of the world radically and call other models into question.
A mathematical discovery doesn’t do that. It might make something more clear, easier to work with, or provide a technique that can be surprisingly applicable elsewhere.
Those are entirely different. Peano developed a system for talking about arithmetic in a formalized way. This allowed people to talk about arithmetic in new ways, but it didn’t show that previous formulations of arithmetic were wrong. Godel then built on that to show the limits of arithmetic, which still didn’t invalidate that which came before.
The development of complex numbers as an extension of the real numbers didn’t make work with the real numbers invalid.
When a new scientific model is developed, it supercedes the old model. The old model might still have use, but it’s now known to not actually fit reality. Relativity showed that Newtowns model of the cosmos was wrong: it didn’t extend it or generalize it, it showed that it was inadequately describing reality. Close for human scale problems but ultimately wrong.
And we already know relativity is wrong because it doesn’t match experimental results in quantum mechanics.
Science is our understanding of reality. Reality doesn’t change, but our understanding does.
Because math is a fundamentally different from science, if you know something is true then it’s always true given the assumptions.
There’s a difference between an advance that repudiates prior understanding and one that doesn’t. You can, in maths - and I assume this is the point - know that you are right, in a way that you can’t with a more… epistemological science. Of course it’s more complex than that, and a lot of maths is pretty sciency, like deriving approximate solutions for PDEs is more experimental than you might imagine, but even though we might make improvements there, we’ll never go ‘oh actually those error bounds are wrong’. They might be non optimal but they’ll never be wrong
This is missing a lot of historical intrigues and “mistakes” in mathematics. Firstly, the way modern mathematical theorems and proofs are built up from axioms is relatively new (a couple hundred years or so). If you go back to Euclid, there are in fact contradictions that can be drawn from his work because he was defining his axioms inappropriately.
In more modern times we have discussions around the “axiom of choice”, and whole fields such as set theory and Fourier analysis faced some major hurdles in just being established.
My point is that math is constantly changing, also on a fundamental level, because new systems and axioms are being introduced. These rarely invalidate old systems, but sometimes they reveal a contradiction in terms that puts limitations on when some system is valid.
This is very similar to when Einstein developed a new framework for describing gravity: It didn’t “disprove” Newton in the sense that Newton’s laws still apply for all practical purposes in a huge range of situations, it just put clearer limits to when they apply and gave a more general explanation to why they apply.
A lot of sciences find core assumptions were not complete or based on the wrong thing. Health practices that have been around for millenia, like like food safety and sanitation, were successfully implemented using the wrong causes because they addressed the real causes. While they were not called science, they still used the same practices of comparing outcomes in the ways available at the time.
Bloodletting was originally to let out evil or something, then was used in formal medicine successfully but the cause it addressed was incorrect. Now we have much better ideas of how and when it helps to make it even more effective, but the underlying reasons and the methods changed completely.
yeah, and physics changes as a science because the actual physics of the universe changes. what are you on about. “we just learn more about it” is pretty much the definition of all sciences.
Math doesn’t change, we just learn more about it.
The mathematical knowledge we had thousands of years ago is still true, and it always will be.
Isn’t that true of almost all the sciences?
The difference is that if something is proven mathematically it’s 100% certain and will not change. In other sciences you may be taught things that later turn out to be flat out wrong.
Bingo, I was taught in genetics class in the 1990s that RNA played a role but DNA was the primary driver and now my understanding is the current consensus is RNA is the primary driver.
When I was growing up, Minnie was the primary Driver, but now the consensus says that it’s Adam.
Not if it’s later shown that your set of axioms lead to a contradiction.
In that case have fun re-proofing everything with new axioms.
Not here to start shit, genuinely curious what people think about Gödel’s incompleteness theorems in relation to us being able to “know” math
Not a mathematician but the way I understand it, is that it merely shows that there are unprovable problems, not that nothing can be proven.
Sounds hella sus now that you mention it 🤔
Not quite. Science is empirical, which means it’s based on experiments and we can observe patterns and try to make sense of them. We can learn that a pattern or our understanding of it is wrong.
Math is inductive, which means that we have a starting point and we expand out from there using rules. It’s not experimental, and conclusions don’t change.
1+1 is always 2. What happens to math is that we uncover new ways of thinking about things that change the rules or underlying assumptions. 1+1 is 10 in base 2. Now we have a new, deeper truth about the relationship between bases and what “two” means.
Science is much more approximate. The geocentric model fit, and then new data made it not fit and the model changed. Same for heliocentrism, Galileos models, Keplers, and Newtons. They weren’t wrong, they were just discovered to not fit observed reality as well as something else.
A scientific discovery can shift our understanding of the world radically and call other models into question.
A mathematical discovery doesn’t do that. It might make something more clear, easier to work with, or provide a technique that can be surprisingly applicable elsewhere.
You’re contradicting yourself.
Is no different than:
Science isn’t changing, our understanding of it is. Same with math.
Those are entirely different. Peano developed a system for talking about arithmetic in a formalized way. This allowed people to talk about arithmetic in new ways, but it didn’t show that previous formulations of arithmetic were wrong. Godel then built on that to show the limits of arithmetic, which still didn’t invalidate that which came before.
The development of complex numbers as an extension of the real numbers didn’t make work with the real numbers invalid.
When a new scientific model is developed, it supercedes the old model. The old model might still have use, but it’s now known to not actually fit reality. Relativity showed that Newtowns model of the cosmos was wrong: it didn’t extend it or generalize it, it showed that it was inadequately describing reality. Close for human scale problems but ultimately wrong.
And we already know relativity is wrong because it doesn’t match experimental results in quantum mechanics.
Science is our understanding of reality. Reality doesn’t change, but our understanding does.
Because math is a fundamentally different from science, if you know something is true then it’s always true given the assumptions.
There’s a difference between an advance that repudiates prior understanding and one that doesn’t. You can, in maths - and I assume this is the point - know that you are right, in a way that you can’t with a more… epistemological science. Of course it’s more complex than that, and a lot of maths is pretty sciency, like deriving approximate solutions for PDEs is more experimental than you might imagine, but even though we might make improvements there, we’ll never go ‘oh actually those error bounds are wrong’. They might be non optimal but they’ll never be wrong
This is missing a lot of historical intrigues and “mistakes” in mathematics. Firstly, the way modern mathematical theorems and proofs are built up from axioms is relatively new (a couple hundred years or so). If you go back to Euclid, there are in fact contradictions that can be drawn from his work because he was defining his axioms inappropriately.
In more modern times we have discussions around the “axiom of choice”, and whole fields such as set theory and Fourier analysis faced some major hurdles in just being established.
My point is that math is constantly changing, also on a fundamental level, because new systems and axioms are being introduced. These rarely invalidate old systems, but sometimes they reveal a contradiction in terms that puts limitations on when some system is valid.
This is very similar to when Einstein developed a new framework for describing gravity: It didn’t “disprove” Newton in the sense that Newton’s laws still apply for all practical purposes in a huge range of situations, it just put clearer limits to when they apply and gave a more general explanation to why they apply.
A lot of sciences find core assumptions were not complete or based on the wrong thing. Health practices that have been around for millenia, like like food safety and sanitation, were successfully implemented using the wrong causes because they addressed the real causes. While they were not called science, they still used the same practices of comparing outcomes in the ways available at the time.
Bloodletting was originally to let out evil or something, then was used in formal medicine successfully but the cause it addressed was incorrect. Now we have much better ideas of how and when it helps to make it even more effective, but the underlying reasons and the methods changed completely.
Bloodletting is still a thing, but it’s called therapeutic phlebotomy.
Source: I have too much iron in my blood so I have to be bloodlet
yeah, and physics changes as a science because the actual physics of the universe changes. what are you on about. “we just learn more about it” is pretty much the definition of all sciences.