Why is that? I've actually looked into this since (completely unexpectedly) discussing it with BB, and I find that the question of whether numbers actually exist outside of a mind (and math, which requires numbers), or are abstract conceptions used by minds to operate in and understand the world, has been debated by philosophers for two millennia with no conclusive answer. Plato said they did but plenty of other philosophers say they don't, and it doesn't seem that those who think they exist independently outside of a mind outnumber those who say they don't.
I think you are somewhat 70% correct about math being created to correspond to physical reality. The nuance is that at first this physical reality created our brains to be able to notice correspondence between objects and formalize them as a thing in itself.
Oh, right, I fully agree with this. I'm definitely not saying that minds come first in the chicken/egg question or that physical reality itself doesn't exist outside of a mind. More that physical reality has resulted in minds, and some of those minds attempt to and are pretty good at understanding, predicting, and manipulating physical reality (to a limited extent), and that math is a critically useful mental tool they use to do so. Something like that.
It seems at least hypothetically possible that there could be an alien species with minds that are completely different from our own, but that are equally or more advanced in technology/reality manipulation, who might use an entirely different type of math that would be entirely incomprehensible to us (but still effective) if their minds worked in a sufficiently different way.
Before talking about aliens and their math, I think its important to notice that even human math is not a single thing. There are all kind of mathematical entities with different behaviours described by different domains of math and what is true for some doesn't have to be true for other. For example, elements of finite fields work differently than natural numbers.
When we say that relity works on math we mean that there are some objects in reality that behave the way that some of the mathematical objects behave. Or, more accurately, statements about mathematical objects are simply a generalized way to talk about some objects in reality which have a particular common property. Instead of making multiple individual statements about all kind of objects:
"One apple plus one apple is two apples"
"One orange plus one orange is two oranges"
"One chair plus one chair is two chairs"
"One sofa plus one sofa is two sofas"
...
We make just one statement:
One of any object that works as natural numbers plus one of the same object that works as natural numbers is two of these objects that work as natural numbers. Or, more laconicly: 1+1 = 2.
Sure, minds capable of understanding mathematical theorems and proofs wouldn't exist if there wasn't a physical reality in which minds could exist, and yes, these minds can use math to understand physical reality---but this does not mean these minds 'created' math to correspond with physical reality. This is literally just wrong. Mathematical theorems are useful in physics, but this is not the reason they are true. We know that the mathematical theorems are true deductively, and we can verify that *some* of them are useful in physics experimentally.
I don't think the second paragraph makes much sense or supports your point at all. Suppose this "different type of math," whatever that means, exists, and suppose it can be used by these aliens to describe physical reality. Why does this imply that our math was created to correspond with physical reality, or that it is less true?
Whether math was created by us as a tool that lets us model and understand reality is different than whether it is "true". I didn't imply that if this alien exists that it makes our math less true, just that different types of math hypothetically might be workable (and "true", to the extent that true means something always works) for different types of minds. It seems *at least possible* that one type of math could be true/always work for us, and a different type could be true/always work for a different alien type of mind.
You posit that math is provably true independent of physical reality, or at least can be proven without it having to correspond to the laws of physics. It seems you also think that if math is true independent of physical reality/laws of physics, then it's also true independent of the minds that discovered it (is that the right word? rather than created?).
That's where I get lost. I hear you saying that math is independently and eternally true and exists somewhere/somehow outside of both physical reality AND abstract thoughts in minds, much like God is posited to be by believers. Now I don't know if you think math is God or proof of God or that you just believe in math like a religious person believes in God.
I am certainly far more inclined to believe the math people telling me that math is true as some sort of non-physical and also non-mental law or thing, because math people have a very good track record of doing things no one else can do and creating technologies and making correct predictions. I'm not inclined to believe religious people when they tell me the same thing about God, because their own track record is terrible at those things. But that's really all I have to go on, in judging either of these claims.
1. Sure, whether math was created as a tool to model physical reality is different from it being true. It isn't created as a tool to model physical reality and it is true. I don't understand what the implication of the alien hypothetical is, and I also don't like it because it relies on us assuming that something that we by definition can't describe or understand exists. I again don't know what 'different type of math' means, you agreed in your other reply that you think basic logic is, like, true. This would mean that 'our' math is also true even if you're an alien with a weird brain that we can't understand or characterize. But again, I really don't like this hypothetical and I'm not sure what it's supposed to demonstrate.
2. I'll reiterate that I'm agnostic regarding whether math 'exists' outside of our minds! I don't know what this means! Why does it seem that if math is true independent of physical reality, that it's also true independent of our minds? Explain!
3. No, you don't keep hearing me saying that, because I've said multiple times that I'm agnostic on whether math exists outside of our minds. I'm a math student, not a philosophy student, I'm not even sure what it would mean for math to exist outside of our minds, I have just said that we can know that math is true without physical observations or experiments. This isn't controversial and literally any mathematician would agree with me.
4. Okay, for the record, most math does not yet have clear applications to physics or engineering. Also, I again don't know what it means for math to be a 'non-mental law,' I have said many times that I don't have a stance on this sort of thing, my position was just that we know that math is true and that it is not created to correspond with physical reality.
I'm really not particularly interested in whether or not numbers 'exist outside of the mind.' The statement you made was made in response to the question: 'if we make up the mathematical laws, it's hard to see why the laws of physics precisely correspond to our mathematical formulations of them.' Your claim that we created math to correspond to physical reality makes some sense in a very specific sense, in that people understand basic mathematical notions, such as those relevant to the axioms we use to construct the number systems, such as distance, completeness (I'm referring to completeness, as in, the completeness property of real numbers), order, etc., with respect to their basic physical intuitions about objects or spaces that they have encountered. But this doesn't mean we created math to correspond with physical reality nor does it mean that any rigorous formulation of these axioms depends on these physical intuitions. I think it would be extremely difficult to conceptualize of any physical reality that would have some property which would complicate or contradict our understanding of the number systems, of basic mathematical axioms, or of mathematical logic. Even if the observable laws of physics were entirely different than they are in our world, the Peano axioms could still be used to construct the natural numbers, and the notion of a natural number would still make sense, and we could still deduce theorems from their basic properties and from basic axioms. The claim that we created math to correspond with physical reality is wrong because mathematicians do not create or prove theorems based on experimental observations from physics about the behavior of physical things. If the observable laws of physics were somehow different, the math that physicists currently use might not be useful to them anymore, but the math would still be true and you could prove that it was true.
> I think it would be extremely difficult to conceptualize of any physical reality that would have some property which would complicate or contradict our understanding of the number systems
Is it the crux? Suppose I give you a more or less coherent description of a universe where nothing works as natural numbers. Would it change your mind about the matter?
> Even if the observable laws of physics were entirely different than they are in our world, the Peano axioms could still be used to construct the natural numbers and the notion of a natural number would still make sense, and we could still deduce theorems from their basic properties and from basic axioms.
And other axioms could be used to construct some other objects and prove theorems about that. The reason we priviledge Peano axioms compared to literally ony other set of axioms is because in our universe there appears to be a lot of physical entities which behave as objects described by Peano axioms. In a universe where nothing behaved like natural numbers, the notion of natural numbers would make much less sense.
You seem to be assuming that if math isn't useful in some physical application that it necessarily 'makes much less sense.' Would you agree that, in a world where no physical situations could be modelled with partial differential equations, we could still prove statements about partial differential equations?
This analogy is stupid because we couldn't logically prove the existence of a winning chess position that is possible without first defining the rules of chess relevant to that position, and thereby 'inventing the game,' or at least inventing enough of the game to be able to prove that that position is both possible and winning.
But---that aside---you're analogy really is not relevant to what I'm talking about at all. The more apt chess analogy for my hypothetical would be "in a world where there weren't any notions of war, conflict, fighting, etc., etc. , we could still define the rules of and play chess." The proposition your analogy is more analogous to is "without knowledge of any ancillary math, we could still find solutions to partial differential equations." This proposition is both false and has nothing to do with the discussion.
You ask 'so what?'. What exactly do you mean? The claim I've been defending is that mathematical theorems can be proven even if they have no physical applications, and that they are not created to be useful in physical applications. This is a true statement. Applications tend to show up long after theorems are proven. Mathematicians (or, at least the vast majority of them), are not working with applications in mind, they are working because they like math. My claim is not about how non-mathematicians decide what math they think is 'important,' which is what I assume you're invoking when you say 'so what'---my claim is about how mathematicians actually go about 'creating math.'
What exactly do you mean by "nothing works as natural numbers?" I don't think any such universe could exist. Independent of the physical qualities of a universe, it would always make sense for there to be quantities such as 1, 2, 3, etc. As long as there is a non-empty universe, our construction of the natural number system would make sense.
But regardless---the point I was making here was that math makes sense regardless of the physical properties of the universe---that no physical facts of the world could contradict math. Even if there was a world in which there weren't things that it made sense to describe with natural numbers (although I don't think this is possible), that would do nothing to contradict our construction of the natural number system, or to contradict any theorems we derive and prove after having constructed that system.
Yes, 'other axioms can be used to construct other objects' obviously, but in a world without things that behaved like natural numbers, the notion of a natural number WOULD still make sense, because we have constructed the natural number system in a logically coherent way.
> What exactly do you mean by "nothing works as natural numbers?"
Exactly that. In our universe when you put an apple to another apple you get two apples. And if you put another, you get three apples. And for any n apples adding another apple to the pile makes it pile of n+1 apples. In the universe where apples do not work like natural numbers something else happens.
> I don't think any such universe could exist.
I would give you an example, but apparently it isn't really our crux of disagreement.
> But regardless---the point I was making here was that math makes sense regardless of the physical properties of the universe---that no physical facts of the world could contradict math. Even if there was a world in which there weren't things that it made sense to describe with natural numbers (although I don't think this is possible), that would do nothing to contradict our construction of the natural number system, or to contradict any theorems we derive and prove after having constructed that system.
Okay, this seems to be purely a semantic disagreement. When you are saying "makes sense" you mean "valid", while I mean a measure of both validness and soundness. Indeed we can agree that reasoning about natural numbers would be valid even in the universe where nothing behaves like natural numbers. But also we can agree that it won't be sound.
Likewise, lets be a bit more precise what we mean by "math". We can be talking about every formal statement about which conclusions follows from which premises - lets call it BIG MATH. Or we can be talking about such statements which premises are actually grounded in our reality - lets call it small math. When Kryptogal says that humans created math to correspond to reality she means the small math - that we've specifically taken the premises that correspond to our reality. When you say that humans did not create math - you mean the BIG MATH. Once again, there is no actual disagreement.
1. I don't think this makes sense, I think objects would always 'work like natural numbers' in any universe. This could only *maybe* not make sense if you stipulate some extremely strange law about a universe, like "whenever an apple is added to a bucket containing one apple, a third apple magically appears in the bucket." But, even if a stipulation of this sort applied to all objects in a wide range of similar circumstances in a given universe, I think that natural numbers would still make sense. The reason that a third apple appeared in the pile would not be that "1+1=3," it would be that this universe has a very strange law that results in the apparition of a third apple whenever there are two. Also, natural numbers could still always be used to count objects, even if the laws that govern piles of objects are very strange. I actually do sort of think this gets to the crux of the disagreement (even though I wasn't sure about this example earlier), which seems to be about whether the applicability of math to the physical laws of the universe in any way determines the validity or soundness of mathematical arguments.
2. No, I think there is disagreement. I don't entirely agree with how you've framed this. I do think that reasoning about natural numbers would be sound in any universe, even if the universe had strange laws about piles of objects. I think that basic mathematical axioms seem to be true independent of facts about physical reality. I thus don't think the 'big math/small math' dichotomy is very accurate or useful way of describing this disagreement.
I can sort of understand how mathematical logic -- or let's just say logic period -- exists independently as a fundamental part of reality, independent of minds. Though that seems somewhat different than "numbers". But I must admit that I likely only think this because logic comes very intuitively to me -- those were the only advanced math classes I could easily grasp and ace, while I struggled with all others -- so I wonder if these are not all ultimately intuitions about what "seems" to each of us to obviously be objectively and independently true/inviolable versus what does not -- whether that's logic or math or god or naturalism etc.
Where you lose me is when you say this: "mathematicians do not create or prove theorems based on experimental observations from physics about the behavior of physical things." I don't fully understand what this means. And this is actually precisely what I was getting at when I asked BB what it meant when there's an unsolved problem in math and how the mathematicians even know that the problem exists or that a solution must exist. His answer just gave an example of a specific such math problem but did not answer my real question and I didn't formulate it very well, so I just dropped it because it was a tangent. But what you have said above points at exactly what I meant. If these theorems do not depend upon the behavior of physical things, and the existence of unsolved math problems also do not depend upon their being some sort of corresponding unsolved problem or paradox in the actual physical world, but only in the realm of abstract mathematical formulas...then?? What are we actually talking about? How do you prove the theorem is true without a corresponding test or example involving physical reality? If it's just a matter of balancing an equation or getting the math to work on its own terms, then how do we know it's not the equivalent of a very difficult and complex semantic game, just using numbers instead of words?
I am admittedly very much not a math person, so forgive me. But something like solving a math-based problem that is very difficult but necessary in order to figure out something like how to get a spaceship to a different galaxy makes sense to me. Solving a very difficult math problem (or proving/disproving a theorem) that exists independently within the realm of math without any necessary correspondence to physical reality does not.
It may very well be the case that when math-brained people say that these theorems exist as independently true and separate from minds -- even if the laws of physics were somehow different(?!) -- that that is true. I'm not saying it's not true, since many of them seem to believe it, and they're the same people who can do things like put spaceships into orbit while others can't, so I'm inclined to believe what they say. But how could I tell? Or be sure that they aren't people who have such a facility with math that the tool has somewhat taken over their minds and wired it such that they start to believe that the math is more real than physical reality (much like a religious person believes that their ideas about God and the supernatural are more "real" than physical reality)?
It's observable to me that sometimes people who very, very good with math start to privilege math above everything else so much that they start to think that reconciling their math is more important reconciling or addressing reality, in a way that leads them to make actual real-life errors or have problems dealing with life. So I can't leave that possibility off the table.
1. I'll start by again saying, I really do not care if numbers 'exist outside of our mind.' In math, we do not just assume that different number systems exist in reality and move on. The number systems are sets that we define using certain axioms, whose elements abide by certain properties that we can prove. Sure, you would not find our constructions of these sets clearly explained in nature, but you could find quantities of things. Clearly there can be 1 a thing, 2 of a thing, 3 of a thing, etc., even if we have not yet stated a rigorous construction of the natural numbers, and even if we know nothing about the physical things themselves, other than that they exist.
2. Mathematics is proven deductively, not experimentally. We start with axioms and we make a series of true statements that logically follow from the axioms to reach theorems. The theorems are just statements or propositions about mathematical objects (numbers, sets, spaces, etc.) that we know deductively to be true following from their definitions and from a series of logical deductions we make. If a theorem couldn't be clearly proven deductively it would not be a mathematical theorem. Higher math is not really about 'balancing equations,' most of the time, it is about making statements that we know to be overwhelmingly and certainly true to arrive at propositions that we know to be overwhelmingly and certainly true. Yes, the statements are about things that are more abstract than physical things, but if we rigorously define these abstract things, then these statements are no less convincingly true (in fact, they are more convincingly true, because there cannot be experimental errors in math). I'm not sure what you mean by 'semantic game' here' I'm honestly not trying to be rude, but I can't understand why you are so confidently skeptical of higher math as a discipline if you admit to not being a 'math person' and thus probably do not have a super accurate understanding of what non-computational math is about. The example Matthew gave to you was perhaps not the best because the proof of Fermat's last theorem is painfully long and complicated, so obviously he couldn't explain it to you on the spot, but if you want to see what simpler proofs look like, they are not hard to find on the internet.
3. I addressed a lot of this just now---the way we prove mathematical theorems without physical experiments is by using deductive arguments.
4. Here's a really basic example that I think anyone can follow. Suppose we could observe a universe in which there were no physical situations, or no situations at all, that could be modelled by quadratic equations. In this world, you could still define a quadratic equation, you could still derive and prove any of its properties, you could still find its roots if they exist, etc. etc. This is because the properties of a quadratic equation can be deduced from its definition. Again, I'm not making claims about anything being separate from our minds, all I'm saying is that math is true independent of its physical applications. If you are seriously asking 'how could I tell,' my answer is 'read a math textbook, preferably one that introduces proofs and proof-writing.' I find the assertion that math is like a religion to be silly. We know that math hasn't 'taken over people's minds' because that is obviously nonsense, research in math is peer-reviewed, and mathematicians prove things, they don't just make assertions.
"We created the math to correspond with physical reality" made me physically wince
Why is that? I've actually looked into this since (completely unexpectedly) discussing it with BB, and I find that the question of whether numbers actually exist outside of a mind (and math, which requires numbers), or are abstract conceptions used by minds to operate in and understand the world, has been debated by philosophers for two millennia with no conclusive answer. Plato said they did but plenty of other philosophers say they don't, and it doesn't seem that those who think they exist independently outside of a mind outnumber those who say they don't.
I think you are somewhat 70% correct about math being created to correspond to physical reality. The nuance is that at first this physical reality created our brains to be able to notice correspondence between objects and formalize them as a thing in itself.
Oh, right, I fully agree with this. I'm definitely not saying that minds come first in the chicken/egg question or that physical reality itself doesn't exist outside of a mind. More that physical reality has resulted in minds, and some of those minds attempt to and are pretty good at understanding, predicting, and manipulating physical reality (to a limited extent), and that math is a critically useful mental tool they use to do so. Something like that.
It seems at least hypothetically possible that there could be an alien species with minds that are completely different from our own, but that are equally or more advanced in technology/reality manipulation, who might use an entirely different type of math that would be entirely incomprehensible to us (but still effective) if their minds worked in a sufficiently different way.
Before talking about aliens and their math, I think its important to notice that even human math is not a single thing. There are all kind of mathematical entities with different behaviours described by different domains of math and what is true for some doesn't have to be true for other. For example, elements of finite fields work differently than natural numbers.
When we say that relity works on math we mean that there are some objects in reality that behave the way that some of the mathematical objects behave. Or, more accurately, statements about mathematical objects are simply a generalized way to talk about some objects in reality which have a particular common property. Instead of making multiple individual statements about all kind of objects:
"One apple plus one apple is two apples"
"One orange plus one orange is two oranges"
"One chair plus one chair is two chairs"
"One sofa plus one sofa is two sofas"
...
We make just one statement:
One of any object that works as natural numbers plus one of the same object that works as natural numbers is two of these objects that work as natural numbers. Or, more laconicly: 1+1 = 2.
Sure, minds capable of understanding mathematical theorems and proofs wouldn't exist if there wasn't a physical reality in which minds could exist, and yes, these minds can use math to understand physical reality---but this does not mean these minds 'created' math to correspond with physical reality. This is literally just wrong. Mathematical theorems are useful in physics, but this is not the reason they are true. We know that the mathematical theorems are true deductively, and we can verify that *some* of them are useful in physics experimentally.
I don't think the second paragraph makes much sense or supports your point at all. Suppose this "different type of math," whatever that means, exists, and suppose it can be used by these aliens to describe physical reality. Why does this imply that our math was created to correspond with physical reality, or that it is less true?
See my other response to you.
Whether math was created by us as a tool that lets us model and understand reality is different than whether it is "true". I didn't imply that if this alien exists that it makes our math less true, just that different types of math hypothetically might be workable (and "true", to the extent that true means something always works) for different types of minds. It seems *at least possible* that one type of math could be true/always work for us, and a different type could be true/always work for a different alien type of mind.
You posit that math is provably true independent of physical reality, or at least can be proven without it having to correspond to the laws of physics. It seems you also think that if math is true independent of physical reality/laws of physics, then it's also true independent of the minds that discovered it (is that the right word? rather than created?).
That's where I get lost. I hear you saying that math is independently and eternally true and exists somewhere/somehow outside of both physical reality AND abstract thoughts in minds, much like God is posited to be by believers. Now I don't know if you think math is God or proof of God or that you just believe in math like a religious person believes in God.
I am certainly far more inclined to believe the math people telling me that math is true as some sort of non-physical and also non-mental law or thing, because math people have a very good track record of doing things no one else can do and creating technologies and making correct predictions. I'm not inclined to believe religious people when they tell me the same thing about God, because their own track record is terrible at those things. But that's really all I have to go on, in judging either of these claims.
Again, I'll respond paragraph by paragraph:
1. Sure, whether math was created as a tool to model physical reality is different from it being true. It isn't created as a tool to model physical reality and it is true. I don't understand what the implication of the alien hypothetical is, and I also don't like it because it relies on us assuming that something that we by definition can't describe or understand exists. I again don't know what 'different type of math' means, you agreed in your other reply that you think basic logic is, like, true. This would mean that 'our' math is also true even if you're an alien with a weird brain that we can't understand or characterize. But again, I really don't like this hypothetical and I'm not sure what it's supposed to demonstrate.
2. I'll reiterate that I'm agnostic regarding whether math 'exists' outside of our minds! I don't know what this means! Why does it seem that if math is true independent of physical reality, that it's also true independent of our minds? Explain!
3. No, you don't keep hearing me saying that, because I've said multiple times that I'm agnostic on whether math exists outside of our minds. I'm a math student, not a philosophy student, I'm not even sure what it would mean for math to exist outside of our minds, I have just said that we can know that math is true without physical observations or experiments. This isn't controversial and literally any mathematician would agree with me.
4. Okay, for the record, most math does not yet have clear applications to physics or engineering. Also, I again don't know what it means for math to be a 'non-mental law,' I have said many times that I don't have a stance on this sort of thing, my position was just that we know that math is true and that it is not created to correspond with physical reality.
I'm really not particularly interested in whether or not numbers 'exist outside of the mind.' The statement you made was made in response to the question: 'if we make up the mathematical laws, it's hard to see why the laws of physics precisely correspond to our mathematical formulations of them.' Your claim that we created math to correspond to physical reality makes some sense in a very specific sense, in that people understand basic mathematical notions, such as those relevant to the axioms we use to construct the number systems, such as distance, completeness (I'm referring to completeness, as in, the completeness property of real numbers), order, etc., with respect to their basic physical intuitions about objects or spaces that they have encountered. But this doesn't mean we created math to correspond with physical reality nor does it mean that any rigorous formulation of these axioms depends on these physical intuitions. I think it would be extremely difficult to conceptualize of any physical reality that would have some property which would complicate or contradict our understanding of the number systems, of basic mathematical axioms, or of mathematical logic. Even if the observable laws of physics were entirely different than they are in our world, the Peano axioms could still be used to construct the natural numbers, and the notion of a natural number would still make sense, and we could still deduce theorems from their basic properties and from basic axioms. The claim that we created math to correspond with physical reality is wrong because mathematicians do not create or prove theorems based on experimental observations from physics about the behavior of physical things. If the observable laws of physics were somehow different, the math that physicists currently use might not be useful to them anymore, but the math would still be true and you could prove that it was true.
> I think it would be extremely difficult to conceptualize of any physical reality that would have some property which would complicate or contradict our understanding of the number systems
Is it the crux? Suppose I give you a more or less coherent description of a universe where nothing works as natural numbers. Would it change your mind about the matter?
> Even if the observable laws of physics were entirely different than they are in our world, the Peano axioms could still be used to construct the natural numbers and the notion of a natural number would still make sense, and we could still deduce theorems from their basic properties and from basic axioms.
And other axioms could be used to construct some other objects and prove theorems about that. The reason we priviledge Peano axioms compared to literally ony other set of axioms is because in our universe there appears to be a lot of physical entities which behave as objects described by Peano axioms. In a universe where nothing behaved like natural numbers, the notion of natural numbers would make much less sense.
You seem to be assuming that if math isn't useful in some physical application that it necessarily 'makes much less sense.' Would you agree that, in a world where no physical situations could be modelled with partial differential equations, we could still prove statements about partial differential equations?
In a world where nobody invented chess, we could still stipulate winning chess board positions. So what?
This analogy is stupid because we couldn't logically prove the existence of a winning chess position that is possible without first defining the rules of chess relevant to that position, and thereby 'inventing the game,' or at least inventing enough of the game to be able to prove that that position is both possible and winning.
But---that aside---you're analogy really is not relevant to what I'm talking about at all. The more apt chess analogy for my hypothetical would be "in a world where there weren't any notions of war, conflict, fighting, etc., etc. , we could still define the rules of and play chess." The proposition your analogy is more analogous to is "without knowledge of any ancillary math, we could still find solutions to partial differential equations." This proposition is both false and has nothing to do with the discussion.
You ask 'so what?'. What exactly do you mean? The claim I've been defending is that mathematical theorems can be proven even if they have no physical applications, and that they are not created to be useful in physical applications. This is a true statement. Applications tend to show up long after theorems are proven. Mathematicians (or, at least the vast majority of them), are not working with applications in mind, they are working because they like math. My claim is not about how non-mathematicians decide what math they think is 'important,' which is what I assume you're invoking when you say 'so what'---my claim is about how mathematicians actually go about 'creating math.'
Just noticed this
What exactly do you mean by "nothing works as natural numbers?" I don't think any such universe could exist. Independent of the physical qualities of a universe, it would always make sense for there to be quantities such as 1, 2, 3, etc. As long as there is a non-empty universe, our construction of the natural number system would make sense.
But regardless---the point I was making here was that math makes sense regardless of the physical properties of the universe---that no physical facts of the world could contradict math. Even if there was a world in which there weren't things that it made sense to describe with natural numbers (although I don't think this is possible), that would do nothing to contradict our construction of the natural number system, or to contradict any theorems we derive and prove after having constructed that system.
Yes, 'other axioms can be used to construct other objects' obviously, but in a world without things that behaved like natural numbers, the notion of a natural number WOULD still make sense, because we have constructed the natural number system in a logically coherent way.
> What exactly do you mean by "nothing works as natural numbers?"
Exactly that. In our universe when you put an apple to another apple you get two apples. And if you put another, you get three apples. And for any n apples adding another apple to the pile makes it pile of n+1 apples. In the universe where apples do not work like natural numbers something else happens.
> I don't think any such universe could exist.
I would give you an example, but apparently it isn't really our crux of disagreement.
> But regardless---the point I was making here was that math makes sense regardless of the physical properties of the universe---that no physical facts of the world could contradict math. Even if there was a world in which there weren't things that it made sense to describe with natural numbers (although I don't think this is possible), that would do nothing to contradict our construction of the natural number system, or to contradict any theorems we derive and prove after having constructed that system.
Okay, this seems to be purely a semantic disagreement. When you are saying "makes sense" you mean "valid", while I mean a measure of both validness and soundness. Indeed we can agree that reasoning about natural numbers would be valid even in the universe where nothing behaves like natural numbers. But also we can agree that it won't be sound.
Likewise, lets be a bit more precise what we mean by "math". We can be talking about every formal statement about which conclusions follows from which premises - lets call it BIG MATH. Or we can be talking about such statements which premises are actually grounded in our reality - lets call it small math. When Kryptogal says that humans created math to correspond to reality she means the small math - that we've specifically taken the premises that correspond to our reality. When you say that humans did not create math - you mean the BIG MATH. Once again, there is no actual disagreement.
1. I don't think this makes sense, I think objects would always 'work like natural numbers' in any universe. This could only *maybe* not make sense if you stipulate some extremely strange law about a universe, like "whenever an apple is added to a bucket containing one apple, a third apple magically appears in the bucket." But, even if a stipulation of this sort applied to all objects in a wide range of similar circumstances in a given universe, I think that natural numbers would still make sense. The reason that a third apple appeared in the pile would not be that "1+1=3," it would be that this universe has a very strange law that results in the apparition of a third apple whenever there are two. Also, natural numbers could still always be used to count objects, even if the laws that govern piles of objects are very strange. I actually do sort of think this gets to the crux of the disagreement (even though I wasn't sure about this example earlier), which seems to be about whether the applicability of math to the physical laws of the universe in any way determines the validity or soundness of mathematical arguments.
2. No, I think there is disagreement. I don't entirely agree with how you've framed this. I do think that reasoning about natural numbers would be sound in any universe, even if the universe had strange laws about piles of objects. I think that basic mathematical axioms seem to be true independent of facts about physical reality. I thus don't think the 'big math/small math' dichotomy is very accurate or useful way of describing this disagreement.
I can sort of understand how mathematical logic -- or let's just say logic period -- exists independently as a fundamental part of reality, independent of minds. Though that seems somewhat different than "numbers". But I must admit that I likely only think this because logic comes very intuitively to me -- those were the only advanced math classes I could easily grasp and ace, while I struggled with all others -- so I wonder if these are not all ultimately intuitions about what "seems" to each of us to obviously be objectively and independently true/inviolable versus what does not -- whether that's logic or math or god or naturalism etc.
Where you lose me is when you say this: "mathematicians do not create or prove theorems based on experimental observations from physics about the behavior of physical things." I don't fully understand what this means. And this is actually precisely what I was getting at when I asked BB what it meant when there's an unsolved problem in math and how the mathematicians even know that the problem exists or that a solution must exist. His answer just gave an example of a specific such math problem but did not answer my real question and I didn't formulate it very well, so I just dropped it because it was a tangent. But what you have said above points at exactly what I meant. If these theorems do not depend upon the behavior of physical things, and the existence of unsolved math problems also do not depend upon their being some sort of corresponding unsolved problem or paradox in the actual physical world, but only in the realm of abstract mathematical formulas...then?? What are we actually talking about? How do you prove the theorem is true without a corresponding test or example involving physical reality? If it's just a matter of balancing an equation or getting the math to work on its own terms, then how do we know it's not the equivalent of a very difficult and complex semantic game, just using numbers instead of words?
I am admittedly very much not a math person, so forgive me. But something like solving a math-based problem that is very difficult but necessary in order to figure out something like how to get a spaceship to a different galaxy makes sense to me. Solving a very difficult math problem (or proving/disproving a theorem) that exists independently within the realm of math without any necessary correspondence to physical reality does not.
It may very well be the case that when math-brained people say that these theorems exist as independently true and separate from minds -- even if the laws of physics were somehow different(?!) -- that that is true. I'm not saying it's not true, since many of them seem to believe it, and they're the same people who can do things like put spaceships into orbit while others can't, so I'm inclined to believe what they say. But how could I tell? Or be sure that they aren't people who have such a facility with math that the tool has somewhat taken over their minds and wired it such that they start to believe that the math is more real than physical reality (much like a religious person believes that their ideas about God and the supernatural are more "real" than physical reality)?
It's observable to me that sometimes people who very, very good with math start to privilege math above everything else so much that they start to think that reconciling their math is more important reconciling or addressing reality, in a way that leads them to make actual real-life errors or have problems dealing with life. So I can't leave that possibility off the table.
I'll respond to this paragraph by paragraph:
1. I'll start by again saying, I really do not care if numbers 'exist outside of our mind.' In math, we do not just assume that different number systems exist in reality and move on. The number systems are sets that we define using certain axioms, whose elements abide by certain properties that we can prove. Sure, you would not find our constructions of these sets clearly explained in nature, but you could find quantities of things. Clearly there can be 1 a thing, 2 of a thing, 3 of a thing, etc., even if we have not yet stated a rigorous construction of the natural numbers, and even if we know nothing about the physical things themselves, other than that they exist.
2. Mathematics is proven deductively, not experimentally. We start with axioms and we make a series of true statements that logically follow from the axioms to reach theorems. The theorems are just statements or propositions about mathematical objects (numbers, sets, spaces, etc.) that we know deductively to be true following from their definitions and from a series of logical deductions we make. If a theorem couldn't be clearly proven deductively it would not be a mathematical theorem. Higher math is not really about 'balancing equations,' most of the time, it is about making statements that we know to be overwhelmingly and certainly true to arrive at propositions that we know to be overwhelmingly and certainly true. Yes, the statements are about things that are more abstract than physical things, but if we rigorously define these abstract things, then these statements are no less convincingly true (in fact, they are more convincingly true, because there cannot be experimental errors in math). I'm not sure what you mean by 'semantic game' here' I'm honestly not trying to be rude, but I can't understand why you are so confidently skeptical of higher math as a discipline if you admit to not being a 'math person' and thus probably do not have a super accurate understanding of what non-computational math is about. The example Matthew gave to you was perhaps not the best because the proof of Fermat's last theorem is painfully long and complicated, so obviously he couldn't explain it to you on the spot, but if you want to see what simpler proofs look like, they are not hard to find on the internet.
3. I addressed a lot of this just now---the way we prove mathematical theorems without physical experiments is by using deductive arguments.
4. Here's a really basic example that I think anyone can follow. Suppose we could observe a universe in which there were no physical situations, or no situations at all, that could be modelled by quadratic equations. In this world, you could still define a quadratic equation, you could still derive and prove any of its properties, you could still find its roots if they exist, etc. etc. This is because the properties of a quadratic equation can be deduced from its definition. Again, I'm not making claims about anything being separate from our minds, all I'm saying is that math is true independent of its physical applications. If you are seriously asking 'how could I tell,' my answer is 'read a math textbook, preferably one that introduces proofs and proof-writing.' I find the assertion that math is like a religion to be silly. We know that math hasn't 'taken over people's minds' because that is obviously nonsense, research in math is peer-reviewed, and mathematicians prove things, they don't just make assertions.
5. I think this is a really silly sort of ad-hom.
Good treatments of the subject which escape youtube
https://archive.philosophersmag.com/the-common-consent-argument/
Fine tuning argument
https://acrobat.adobe.com/id/urn:aaid:sc:VA6C2:4f8e91d3-dadf-4d40-8c1b-c9e4de25b0d3