Mathematical and logical truths exist before we have discovered them, so

  • #51
Pythagorean
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We've already established that the QM argument is irrelevant (I feel like this is the fifth post I've said it, really). You've asserted that it doesn't apply to you, but you keep trying to defend it as if it does.

You keep making these statements about the human brain. First you said that there's some essential part of mathematics that is connected to human brains having evolved in the macroscopic world but I think I've demonstrated that our mathematics works just fine on the quantum level and nothing is invalidated by QM nor has to be corrected for that.
You're tying two merge arguments together here, once again bringing QM back into it. If you're not arguing the fundamental nature of the universe is mathematical, then this argument is dead!

The technique of mathematics can be applied to anything, that doesn't make it invented. It is with our brains that we apply the technique of mathematics, that is the argument. Ironically, having an answer for everything is a symptom of pseudoscience (but not sufficient to call mathematics pseudoscience, of course). You give me something, and I guarantee I'll be able to use my creativity and fit mathematics to any question you ask me. That's not because mathematics is discovered.

With many inventions, the full capabilities of the invention are discovered later. The way the invention can be used is discovered, but the invention is invented.

And I still assert that your desire for mathematics to be discovered is a consequence of your brain developing in a macroscopic world (I meant this in terms of your sensory and intellectual experience, by the way). And I did present evidence (don't know if you watched the lectures or not from that previous thread way back when) about how the brain sees patterns that aren't there. It's a common psychological experiment that can be proven to a whole audience.

Is there anything in particular you can point to that would indicate what about our mathematics would be incomparable or incompatible with some other sort of mathematics? Or anything you can point to in mathematics that seems particularly dependent upon the human brain for it to be true?
How could we tell and how would that prove mathematics is discovered? The whole point is that we're a part of the universe, interacting with it, not something safely studying it from behind a glass window. We change the universe by seeing it how we want to (please don't interpret this as mind over matter will power bs, think more practically like when we imagine a space shuttle in the sky, then we send it there, interacting with and changing our universe)

no, I can't prove that mathematics is invented. I can only debunk your claims that mathematics is discovered.

The whole question may very well be meaningless in the context of dualism (which is what we're practicing here)

Again, if they were presented with a definition of π, would they be unable to see how it had anything to do with circles or trigonometry or wave mechanics or whatever analogs of those things might exist in their mathematics? Would they conclude somehow that we had reached an incorrect value for π?

And does this all apply to their physics too? Would our physics seem to them like a bunch of nonsense unconnected to the physical world?

I don't think so. I think it would be just like I said above:

And similarly, I think that π and its relationship to circles, trigonometric and other periodic functions, and wave mechanics is a mathematical fact that could not contradict anything within an alien's mathematics.
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Ok, still, we can invalidate the crappy hypothetical alien argument with the fact that people invent things independently all the time. Just because a human and an alien both invent the same theories and axioms, does not mean they are discovered. Furthermore, it shouldn't imply that they're correct! There's no way for you to tell if the axioms you chose are cutting you off from a deeper understanding of reality.

And no, their interactions may not be the same as ours at all, there's no way to guarantee they would experience the world the same us. We could drop them off a skyscraper and measure the physics of their collisions with the ground and we'd interpret physics the same as we always did, but there's no reason they'd develop the same technological/scientific perception of the universe that we have, unless they're brains are a lot like ours, in which case we're not surprised when multiple people from the same race come up with different inventions, so it shouldn't be much of a stretch for a similar neurological composition.
 
  • #52
Pythagorean
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I do not think that's true. I think that the sum of a triangle's angles being 180° is either a fact that exists independently from any intellect thinking about it, or a fact based upon some more fundamental mathematical relationships that exist independently from any intellect thinking about them. And hence the sum of 180° is something that could be reached by different humans not communicating with each other and even by aliens as well.
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How can you prove that a triangles angles even exist without a human thinking about it? There's no triangles in reality. We invented the triangle to address triangle-like things in reality.

But if you really consider even physics and the subject it studies to be wholly invented, an exercise in human navel-gazing as it were, as invented as Swedish grammar or the grammar of human languages in general†, then yeah, I guess there's nothing to talk about.
Actually, I said the thing physics studies is actually discovered. Remember in my tool analogy that the chainsaw is cutting the tree. The tree is discovered. The chainsaw is invented. Physics is invented.

And mathematics studies itself as far as I can tell. You'll still have to tell me more about the other elusive subject of mathematics.
 
  • #53
We've already established that the QM argument is irrelevant (I feel like this is the fifth post I've said it, really). You've asserted that it doesn't apply to you, but you keep trying to defend it as if it does.



You're tying two merge arguments together here, once again bringing QM back into it. If you're not arguing the fundamental nature of the universe is mathematical, then this argument is dead!
Uh, okay... but it was only a few comments back where, in that bit I quoted, you appeared to be claiming that in some context some part of mathematics stops working. And you have continued making various assertions related to QM in this most recent response.

So does what you're saying above mean that you're no longer advancing the bit about something that stops mathematics from working in a particular frame, and we can agree that there isn't any context in physics where mathematics stops working?

And I still assert that your desire for mathematics to be discovered is a consequence of your brain developing in a macroscopic world (I meant this in terms of your sensory and intellectual experience, by the way). And I did present evidence (don't know if you watched the lectures or not from that previous thread way back when) about how the brain sees patterns that aren't there. It's a common psychological experiment that can be proven to a whole audience.
I may or may not have, I don't remember. If it demonstrated something that would show that some part of mathematics was untrue or no longer works outside of the context of a human brain, I would be impressed. If all it shows is that the human brain sometimes interpolates parts of the patterns it sees, that's really nothing like proving that mathematics is dependent on the human brain and wouldn't work in the absence of it.

Ok, still, we can invalidate the crappy hypothetical alien argument with the fact that people invent things independently all the time. Just because a human and an alien both invent the same theories and axioms, does not mean they are discovered. Furthermore, it shouldn't imply that they're correct! There's no way for you to tell if the axioms you chose are cutting you off from a deeper understanding of reality.
Okay, you seem to be saying here that if aliens and humans began from the same axioms they would reach the same conclusions, which I what I'm saying. As I've emphasized, I'm not saying anything about any particular set of axioms or any particular formulation of mathematics. I'm talking about the thing that forces a specific relationship between any given set of axioms and any particular conclusion.

How can you prove that a triangles angles even exist without a human thinking about it? There's no triangles in reality. We invented the triangle to address triangle-like things in reality.
I'm not saying anything like "triangles are real".

Actually, I said the thing physics studies is actually discovered. Remember in my tool analogy that the chainsaw is cutting the tree. The tree is discovered. The chainsaw is invented. Physics is invented.
Okay - so just to be clear here, in response to that direct question above are you saying that you think that both physics and the object of its study are no more related to an underlying reality, than grammar is and are both studying a purely human-derived topic?

And mathematics studies itself as far as I can tell. You'll still have to tell me more about the other elusive subject of mathematics.
You seem to be saying above that, beginning from the same axioms, both humans and aliens would reach the same conclusions. That is what I am talking about. Whatever part of what mathematics studies that would constrain the conclusions of both humans and aliens can't be something that derives purely from humanity.
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  • #54
Pythagorean
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Uh, okay... but it was only a few comments back where, in that bit I quoted, you appeared to be claiming that in some context some part of mathematics stops working. And you have continued making various assertions related to QM in this most recent response.
I'm not claiming math stops working. Work (in all definitions of the word) does not signify discovery or invention. It's irrelevant to this argument. I was simply saying there's no panacea in mathematics, to which you concurred and that argument is over. The specific form to which you concurred was "the universe is not fundamentally mathematical".

So does what you're saying above mean that you're no longer advancing the bit about something that stops mathematics from working in a particular frame, and we can agree that there isn't any context in physics where mathematics stops working?
I never stated that. I think this part of your bias against physicists (which you've admitted to having, then criticized me for claiming after admitting, by the way, looking back).

This is part of the same "no panacea" argument. I didn't say mathematics itself stops working. The particular math you're using for an interaction does not fit the observations for all cases. In some cases, the particular math you're using is completely wrong if one parameter gets to small (for instance), so you can only use one math for once case, and another math for another case. (The classic examples of this arise in QM vs. Classical Physics, which is why it came up, nobodies claiming that all of mathematics stops working.)

This may not have to do with the discovered/invented argument, but the fact that you keep interpreting that way shows a fundamental difference in how we view math. You view all your knowledge of mathematics as one, intertwined thing, but I view mathematics as a whole bunch of different random things. I should be using the word tools for mathematics, rather than tool.

I
may or may not have, I don't remember. If it demonstrated something that would show that some part of mathematics was untrue or no longer works outside of the context of a human brain, I would be impressed. If all it shows is that the human brain sometimes interpolates parts of the patterns it sees, that's really nothing like proving that mathematics is dependent on the human brain and wouldn't work in the absence of it.
That's not what I'm claiming, though. That's your straw man, and its distracted from my argument quite a bit. I'm simply claiming there's no way for you to show mathematics is independent of the mind. I'm suggesting that perhaps the notion comes from your mind.

Okay, you seem to be saying here that if aliens and humans began from the same axioms they would reach the same conclusions, which I what I'm saying.
Not in general. Like I said, aliens might not think like us at all, they might not even "think" in the traditional sense. Many consider insects to lack thought (no brain, bundle of nerves). There's no guarantee that any other species uses mathematics besides humans.
I'm not saying anything like "triangles are real".
real/not real isn't the discussion. You're claiming triangles are discovered (or at least that the angles between them are, which would be silly if you admitted the triangle was invented).


Okay - so just to be clear here, in response to that direct question above are you saying that you think that both physics and the object of its study are no more related to an underlying reality, than grammar is and are both studying a purely human-derived topic?
Why do you keep planting that? No, I didn't say that the object that physics studies is not related to reality. Physical reality is the subject of physics. It is reality, I'm just avoiding saying that it's all of reality by calling it physical reality.

You seem to be saying above that, beginning from the same axioms, both humans and aliens would reach the same conclusions. That is what I am talking about. Whatever part of what mathematics studies that would constrain the conclusions of both humans and aliens can't be something that derives purely from humanity.
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I'm saying it's possible (given that the particular aliens are remarkably similar to humans), but I'm also saying it's not an argument for discovery.
 
  • #55
I'm not claiming math stops working. Work (in all definitions of the word) does not signify discovery or invention. It's irrelevant to this argument. I was simply saying there's no panacea in mathematics, to which you concurred and that argument is over. The specific form to which you concurred was "the universe is not fundamentally mathematical".



I never stated that. I think this part of your bias against physicists (which you've admitted to having, then criticized me for claiming after admitting, by the way, looking back).
Okay, I'm going to again quote what I was talking about:
The argument with QM is that we have to change frames. One math works in one frame, the other math works in another frame, and even then, you're still generalizing.
If "One math works in one frame, the other math works in another frame" is not conveying some sort of incompatibility between some part of mathematics and a particular frame (like QM, you appear to be indicating in the first sentence) I don't know what it's expressing but I don't think it's somehow wildly unreasonable and prejudiced for me to interpret it that way.

It isn't my imagination or some sort of bias that makes me think that you and turbo and others say things like this. You actually say such things.
This is part of the same "no panacea" argument. I didn't say mathematics itself stops working. The particular math you're using for an interaction does not fit the observations for all cases. In some cases, the particular math you're using is completely wrong if one parameter gets to small (for instance), so you can only use one math for once case, and another math for another case. (The classic examples of this arise in QM vs. Classical Physics, which is why it came up, nobodies claiming that all of mathematics stops working.)
Okay, so we're again back to a complaint about the nature of mathematics that is based upon its utility to physicists and difficulties that physicists encounter when trying to apply mathematics.

We've now come full circle back to the first thing I was saying in this thread, "But physicists erroneously applying mathematics to a problem of science isn't the same thing as a flaw or limitation in mathematics itself" which you [post=1980511]quoted and asserted was bias[/post].

The claims I make about the sort of things that physicists say, which you keep claiming is some sort of biased view or assumption on my part, you actually keep repeatedly saying.

And these weren't just some off-the-cuff statements about mathematics as a tool, according to you "I think what me and turbo are both trying to do is preemtively dismantle the argument that some mathematicians make" [post=1988054]^[/post], an argument you thought I was making about the relationship between mathematics and reality. So you can't dismiss this now as some statement I misunderstood because of my supposed bias and prejudice - by your own characterization, you were putting forward this argument based upon the utility of mathematics to physicists as some sort of reflection upon the fundamental nature of mathematics.

I understand that you were mistaken in what you thought I was saying but that doesn't mean that I'm somehow biased - I am describing the way that you and (some) other physicists do actually behave. When it serves your purpose, in discussing the fundamental nature of mathematics you will very readily blur the line between mathematics and applied mathematics within physics, and imply that some problem that physicists run into in applying mathematics extends to being a general problem with mathematics itself.
This may not have to do with the discovered/invented argument, but the fact that you keep interpreting that way shows a fundamental difference in how we view math. You view all your knowledge of mathematics as one, intertwined thing, but I view mathematics as a whole bunch of different random things. I should be using the word tools for mathematics, rather than tool.
You may have not covered this in your studies so far but pretty much all of mathematics is intertwined. There are isomorphisms all over the place between widely disparate parts of mathematics. An example in computer science that is most prominent and frequently-encountered is the equivalence between any regular grammar (a mathematical, not a linguistic grammar) and its corresponding Turing machine.
That's not what I'm claiming, though. That's your straw man, and its distracted from my argument quite a bit. I'm simply claiming there's no way for you to show mathematics is independent of the mind. I'm suggesting that perhaps the notion comes from your mind.
Actually, you have very firmly and definitively stated on several occasions that mathematics derives from the human brain - no "suggesting" about it.

There isn't any "straw man" involved - you said that these videos are evidence for the connection between mathematics and particularities of the human brain. I'm not misrepresenting your argument at all, I'm saying that what you claim as evidence for it is probably simply something about pattern matching mechanics in the brain.

From your description it doesn't sound like they demonstrate any reason why mathematics cannot be independent of the mind, nor furnish evidence that would show that mathematics is dependent on the human mind. If all they do is talk about the mechanisms for pattern-matching in the human brain I'm not going to bother to track down the links and watch them, I have read quite a bit about that.
Not in general. Like I said, aliens might not think like us at all, they might not even "think" in the traditional sense. Many consider insects to lack thought (no brain, bundle of nerves). There's no guarantee that any other species uses mathematics besides humans.
Okay, but would you agree that any intelligence that could formulate and understand the same set of axioms, whether from a human mathematics or from the mathematics of a hive-mind insectoid species of aliens or something exotic like that, would reach the same conclusions? It doesn't matter whether they actually exist or not.

The hypothetical is about whether an intelligence unrelated to humans, in comprehending a set of mathematical rules, could reach a conclusion incompatible with human mathematics. Could they end up concluding a different value for π, is the example I have been using.

If they would not arrive at a different value for π then obviously there is something independent of humanity that is placing a constraint on the reasoning surrounding at least that part of mathematics. And this is the same reason why independent groups of humans also arrive at the same value for π, not some aspect of the human brain.
real/not real isn't the discussion. You're claiming triangles are discovered (or at least that the angles between them are, which would be silly if you admitted the triangle was invented).
No, I am not saying that triangles or the angles between them are discovered. As I've emphasized repeatedly, exactly what I have been asserting is discovered about mathematics is extremely abstract - I'm talking about the relationships that underlie mathematics, the things that cause the isomorphisms. The only specific thing I've gotten anywhere close to calling discovered is the ratio π. I have consistently said that much of the description and symbology and formulation of human mathematics is human. (And I think you know very well that "discovered" and "real" are the same concept in this discussion, certainly at least as far as the way I've been using them... you responded quite readily to my questions about physics being related to reality below.)
Why do you keep planting that? No, I didn't say that the object that physics studies is not related to reality. Physical reality is the subject of physics. It is reality, I'm just avoiding saying that it's all of reality by calling it physical reality.
OKAY, in that case you know exactly what I have meant by saying that mathematics is related to reality rather than being a human invention, and why I have been drawing all of these parallels between mathematics and physics, and you can stop pretending that you don't know what I mean.

I have been proposing that the relationship between mathematics and reality is the same as the relationship between physics and reality and we can stop quibbling about the technical meanings of "tool" and "discovered" and "invented" and "real" and "independent of humanity" whether or not you technically characterized or did not characterize physics as one or the other because I think that it's clear you know exactly what I'm talking about when it's physics that is the topic of discussion.
I'm saying it's possible (given that the particular aliens are remarkably similar to humans), but I'm also saying it's not an argument for discovery.
Whatever! Unless you're going to say that it would be possible for some non-human intelligence to begin from the same set of axioms and reach a different conclusion from humans, reach something like a different value for π, then there is something that is independent of humanity that is part of the makeup of the subject studied by mathematics.
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  • #56
Pythagorean
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Okay, so we're again back to a complaint about the nature of mathematics that is based upon its utility to physicists and difficulties that physicists encounter when trying to apply mathematics.

We've now come full circle back to the first thing I was saying in this thread, "But physicists erroneously applying mathematics to a problem of science isn't the same thing as a flaw or limitation in mathematics itself" which you [post=1980511]quoted and asserted was bias[/post].
Because you're assuming it's an erroneous application. Why is it so hard to believe that there is no universally correct method or that there will always be uncertainty involved with observing reality (especially when your observations are also interactions). My argument isn't about the failure and impotence of math, but about the elusiveness of reality.

The claims I make about the sort of things that physicists say, which you keep claiming is some sort of biased view or assumption on my part, you actually keep repeatedly saying.
You interpret it differently though and go off-track with your responses. You've been arguing about the math working in all frames when I never claimed that all of math somehow breaks down (which would be a strange concept to me, since you don't apply all mathematics at once). You want somebody to be at fault, or be guilty for it and you don't want it to be math. I don't want it to be math OR physics! I think it's the limitations of human thinking that are guilty for the limitations of human understanding. Yet, you keep assuming that I'm on physics side and you're on math's side. I'm on the philosopher's side, here. But rather than arguing about how we argue, it might be best to start over with premises and conclusions (something we never did in the last thread). Of course, you should start since you're the affirmer.

And these weren't just some off-the-cuff statements about mathematics as a tool, according to you "I think what me and turbo are both trying to do is preemtively dismantle the argument that some mathematicians make" [post=1988054]^[/post], an argument you thought I was making about the relationship between mathematics and reality. So you can't dismiss this now as some statement I misunderstood because of my supposed bias and prejudice - by your own characterization, you were putting forward this argument based upon the utility of mathematics to physicists as some sort of reflection upon the fundamental nature of mathematics.
I sincerely thought you were making the claim about the universe being fundamentally mathematical, and you did admit that you liked that point of view and that you were tempted to argue it, so perhaps it slips through your posting, or perhaps turbo and I were just imagining things (perhaps turbo has a different perspective alltogether, I shouldn't really be speaking for him)

I understand that you were mistaken in what you thought I was saying but that doesn't mean that I'm somehow biased - I am describing the way that you and (some) other physicists do actually behave. When it serves your purpose, in discussing the fundamental nature of mathematics you will very readily blur the line between mathematics and applied mathematics within physics, and imply that some problem that physicists run into in applying mathematics extends to being a general problem with mathematics itself.
Your bias comes through your choice of supporting language, not so much your points, which makes it hard to confront (kind of like someone who's being passive-aggresive). This is why I bolded specific words when I quoted you in this thread; I didn't bold your whole point. I thought it was obvious from what I chose to bold that I wasn't arguing your point. (This is also what I meant about trolling).

On the other hand, it could be imagination, as I've said before, but that doesn't mean I can easily forget about it.

You may have not covered this in your studies so far but pretty much all of mathematics is intertwined. There are isomorphisms all over the place between widely disparate parts of mathematics. An example in computer science that is most prominent and frequently-encountered is the equivalence between any regular grammar (a mathematical, not a linguistic grammar) and its corresponding Turing machine.
No, and there's no guarantee I ever will. I don't get to see behind the scenes of math in my curriculum.

So you'll have to prove that claim. Intertwined is too creative of a word though. Let me be more specific about what I meant: show me that you can get to any other mathematical principal from one mathematical principal (or where a mathematician claims it). I just don't see how topology has anything to do with the Hilbert space (but I'm completely open to any proofs).

Of course, this still isn't an argument for discovery so much as for the elegance and complexity of mathematics. It could still be invented or discovered.

Actually, you have very firmly and definitively stated on several occasions that mathematics derives from the human brain - no "suggesting" about it.
These are simply my observations. Do you deny that mathematics come from the brain (even if you may not agree that it's an argument against discovery?) The only place I've ever seen mathematics is when humans use it, you can't deny that. The "suggestion" is that it's invented.

On the other hand, you claim to know a deeper truth based on no observations, only thought (never has a circle or pi, or a triangle been observed. They are ideals).

There isn't any "straw man" involved - you said that these videos are evidence for the connection between mathematics and particularities of the human brain. I'm not misrepresenting your argument at all, I'm saying that what you claim as evidence for it is probably simply something about pattern matching mechanics in the brain.
No, I said the videos are evidence that the brain sees patterns when none or there.

The fact the brain see patterns that aren't their is not evidence is suggestive of invention.

Okay, but would you agree that any intelligence that could formulate and understand the same set of axioms, whether from a human mathematics or from the mathematics of a hive-mind insectoid species of aliens or something exotic like that, would reach the same conclusions? It doesn't matter whether they actually exist or not.
Yes, I do. Just like you say: "any intelligence that could formulate and understand the same set of axioms", well then of course they'd reach the same conclusions. But that's assuming the conclusion isn't it? Or rather, you're asking me to assume the conclusion that is subject to debate. The axioms are the invention/discovery.

The hypothetical is about whether an intelligence unrelated to humans, in comprehending a set of mathematical rules, could reach a conclusion incompatible with human mathematics. Could they end up concluding a different value for π, is the example I have been using.
You'd still have to show me how this proves your conclusion.

If they would not arrive at a different value for π then obviously there is something independent of humanity that is placing a constraint on the reasoning surrounding at least that part of mathematics. And this is the same reason why independent groups of humans also arrive at the same value for π, not some aspect of the human brain.
You use the word obviously like I should just accept your argument without a proof.



I really think we should start over though, now that we're more familiar with each others vocabulary and the types of semantics involved, and I will simply play skeptic and try to avoid asserting that mathematics is invented (I really am in the middle, and my intuition wants to agree with you because it would be a beautiful thing and the universe would be a wonderful place, but I still don't see it in terms of logic)

Give me your argument in this form.


Premise 1:
...
Premise n-1:
Premise n:
Conclusion: all of mathematics is discovered
 
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  • #57
Because you're assuming it's an erroneous application. Why is it so hard to believe that there is no universally correct method or that there will always be uncertainty involved with observing reality (especially when your observations are also interactions). My argument isn't about the failure and impotence of math, but about the elusiveness of reality.
...and you only say that because you assume that mathematics has nothing to do with reality! Sheesh, and you want to label me as biased!

When physicists tried to apply the mathematics they used in classical physics at the quantum level, that was an erroneous application of mathematics. But despite the context in which you brought it up, it has no bearing on whether or not mathematics is discovered nor invented. Nor even were someone proposing that there's some matrix-like mathematical substrate to the universe, would it have anything to do with that.

You're simply mincing words here again. I said nothing remotely like "there is a universally correct method in physics" all I have done is accurately describe how you behave in projecting difficulties physicists encounter as some sort of reflection on mathematics and upon what connection mathematics may have to reality. That's exactly what you have done, repeatedly. Trying to cast an accurate description of your behavior as some kind of bias is purely rhetorical.

(Why would you have started talking about reality being elusive all of a sudden, anyways? That would not be a direct response to anything I've said.)
Yet, you keep assuming that I'm on physics side and you're on math's side. I'm on the philosopher's side, here.
You're the one doing all the assuming here. You're the one declaring that I must be biased, no matter whether you behave in exactly the manner I describe, and who keeps declaring your own viewpoint as the "middle view" or as being "on the philosopher's side."

If you were really on the philosopher's side you wouldn't have to keep declaring me biased or playing around with the meaning of "discovered" and "invented" and "reality" to pretend you don't understand what I'm talking about and you would have faced head-on the analogies I posed between the degree of reality represented by G and π.
I sincerely thought you were making the claim about the universe being fundamentally mathematical, and you did admit that you liked that point of view and that you were tempted to argue it, so perhaps it slips through your posting, or perhaps turbo and I were just imagining things
Or perhaps I mean what I'm actually, literally saying and you don't need to level accusations of bias at me that you can't demonstrate based upon guesses about my intentions. It would be one thing if you were successful at doing that, but so far you haven't been.

And for the record, turbo responded to me saying "...there aren't any civilizations that have concluded that 2 + 2 = 5 that I know of." with the statement "It seems that our mathematics cannot be used to construct reasonable models of the quantum world." - nothing even remotely similar to me postulating some Matrix-like mathematical substrate to reality.
So you'll have to prove that claim. Intertwined is too creative of a word though. Let me be more specific about what I meant: show me that you can get to any other mathematical principal from one mathematical principal (or where a mathematician claims it). I just don't see how topology has anything to do with the Hilbert space (but I'm completely open to any proofs).
No. The interconnectedness of mathematics doesn't have anything to do with the points I'm trying to make about mathematics, it was simply a response to you attempting to imply that my understanding of mathematics is flawed. If you want to try to prove that mathematics is all disjointed and unrelated, you can go ahead and do that.
These are simply my observations. Do you deny that mathematics come from the brain (even if you may not agree that it's an argument against discovery?) The only place I've ever seen mathematics is when humans use it, you can't deny that. The "suggestion" is that it's invented.

On the other hand, you claim to know a deeper truth based on no observations, only thought (never has a circle or pi, or a triangle been observed. They are ideals).
π, as I have said again and again and again, is no less real that the universal gravitational constant. It exists in reality just as much as G does. How about you respond to that for once instead of tapdancing around it and pretending as though I've claimed triangles are real / discovered, when I explicitly said that I have not?
Yes, I do. Just like you say: "any intelligence that could formulate and understand the same set of axioms", well then of course they'd reach the same conclusions. But that's assuming the conclusion isn't it? Or rather, you're asking me to assume the conclusion that is subject to debate. The axioms are the invention/discovery.
(bolding mine) In fact that is not assuming my conclusions... I asked you if you agreed with that, rather than asking you to assume it for the sake of argument. And you've just finally signified that you agree with it rather than squirming around trying to avoid the issue.
Give me your argument in this form.


Premise 1:
...
Premise n-1:
Premise n:
Conclusion: all of mathematics is discovered
Funny how you slip "all of mathematics" in there as the conclusion when I have repeatedly specified that it's a very abstract part of the subject mathematics studies that I'm saying is discovered. But who am I to point out sketchy tricks of sophistry you're pulling, you're the one on the side of philosophy, after all.

Aaaanyways...

Premise 1: As you yourself affirm, we agree that any intelligence, even an intelligence entirely unlike that embodied by the human brain, which could formulate and understand a particular set of mathematical axioms would arrive at the same conclusions humans arrive at based upon those axioms.

If Premise 1 is true independent of the point in history such a hypothetical scenario might occur at then in the absence of humankind, before its existence or after it perishes, the same alien intelligences would arrive at the same conclusions based upon the same axioms.

Ergo, the relationship between the set of mathematical axioms and the conclusion is a constraint that is independent of the existence of humans or human brains or human-like brains.

If something is independent of the existence of humans it would not be characterized as invented by humans; hence it would fall within the opposite of "invented", which by the conventions of this discussion has been designated "discovered".

the relationship between the set of mathematical axioms and the conclusion is a constraint that is discovered rather than invented.​
And note that again this does not depend on any aliens actually existing. I am demonstrating that based upon a statement that you agreed with, in the way you yourself view mathematics some part of it has existence independent from humans.
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  • #58
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So does that mean that a world can exist through purely a system of numbers, values and rules?

This system can exist without any physical matter, as it is only a mathematical pattern/system, and not an object.

The system can include such things as "time" and "dimensions", but only the mathematical interpretations behind them. As this system of rules and patterns evolves to deeper and deeper complexities, things resembling 'life' and 'objects' can exist. (It is still only the values and properties of these things that exist, and not the objects themselves).

I propose that humans and the universe are nothing more than one part of a mathematical system that can and always has existed without necessary "existing" any more than the number 4 'exists'.

This explains why there is no real analogy or familiarity to explain the phenomena in the quantum world, the particles and fields really are nothing more than values and numerical properties that follow rules.

This also means that every other possible (stable) system of values and rules does exist just as much as ours does (which links in with the multiple universes idea), which explains how life originated despite the improbabilities.

Furthermore, I think that if something is possible, then it has to 'exist', just because we are nothing but a set of values following a set of rules.

Tell me your thoughts, I haven't had too long to think about it, I just wanted to here someone else's view.
Gödel, I and the most of others here (I think) disagree.

Our truths and numbers do not hang out there like birds on the tree. It is not that there are not “things out there”. Universe does not disappear when we close our eyes, but it does need us to give our own meanings (truths, numbers etc.) to it.

Take it as there are patterns out there, like on rock displaying layers of sediments. Few centuries ago these patterns had no meaning other than let’s say decorative. Today we are assigning quite a lot of meanings to them. And we might read more stories from them tomorrow and correct some we are telling now.

Kind regards,
 
  • #59
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Gödel, I and the most of others here (I think) disagree.

Our truths and numbers do not hang out there like birds on the tree. It is not that there are not “things out there”. Universe does not disappear when we close our eyes, but it does need us to give our own meanings (truths, numbers etc.) to it.

Take it as there are patterns out there, like on rock displaying layers of sediments. Few centuries ago these patterns had no meaning other than let’s say decorative. Today we are assigning quite a lot of meanings to them. And we might read more stories from them tomorrow and correct some we are telling now.

Kind regards,
Hi, I don't know if you read my more recent post, which is slightly different to my origional idea, here it is:

Our universe follows rules, these rules (as far as we can see) have always been obeyed. All of these rules so far can be defined through mathematics, so I presume that the fundamental rules of the universe are 'written' in a language similar to maths.

The place where maths must differ from the fundamental code of the universe is that in maths, one particular number can be used in a multitude of contexts. This inherent ambiguity of the different numbers means that the system of our universe must not be written in just maths. So far, the tool of maths has been sufficient in describing and predicting the ways of our universe, as it is easy to tell someone the necessary interpretation behind your numbers.

Dimensions, Time, objects and even emotions are just our own interpretations of the different patterns, phenomena, and types of values that exists in this fundamental system. For example, we interpret one type of 4 to mean distance, another type of 4 to mean electronic charge, and a third type of 4 to perhaps mean a distance in time. Maths does not distinguish between these fours, but the system of our universe does.

Other types of system that cannot be described using maths, and are completely different to the system of our own universe can exist, and may hold other amazing phenomena (like life in ours) that is so separate to our system that we can not imagine it. This endless amount of systems makes it less amazing that life originated, and perhaps makes something as unlikely as the origin of life not unlikely at all.

Do you still disagree?

George
 
  • #60
Most of my discussion here has been with Pythagorean, re-hashing this old argument we had from earlier this year, but I thought I'd mention as a side note: in all the thinking I've done about this sort of thing, I have come to wonder how essential the concept of numbers themselves are to the rest of mathematics.

In trying to imagine how bizarre and remote a perspective some alien mathematician might have in looking at human mathematics, with input from some of the notions that early human civilizations had, it has occurred to me that perhaps you could have a system of mathematics in which it's difficult or impossible to conceive of numbers greater than 1.

It seems to me that, coming from such a perspective, you could still have things like π by regarding them as ratios between small fractions of 1.

I ought to crack open some of my old college books and see whether it's possible to arrive at some system similar to real and complex numbers if you start from numbers needing to have absolute values smaller than 1...
⚛
 
  • #61
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Hi, I don't know if you read my more recent post, which is slightly different to my origional idea, here it is:

Our universe follows rules, these rules (as far as we can see) have always been obeyed. All of these rules so far can be defined through mathematics, so I presume that the fundamental rules of the universe are 'written' in a language similar to maths.
You are not recognizing a fundamental assumption you hold as true without reason, in order to claim this.
Humans are made of the stuff we are trying to define with the language of mathematics.
As Carl Segan said, "we are space dust".
This means we are the universe trying to understand itself.
The meaning we attribute to observation is perhaps more thorough than the meaning
a turtle might attribute to the same event, but it is still the meaning arising from the
reasoning of humans.
Turtles could have the discussion we are having now and be just as certain that the
universe "follows" their rules or reason, and none of us could dispute it unless we
could reason their way and find fault with their axioms.
Our axioms are what "ALL" of mathematics must stand on, and are considered "self evident".
Think about that term, does that "self" mean you, me, all of us humans or the universe?

What is "self evident" to turtles MUST by definition be true reason and will be as
true to turtles as ours are to us. The universe will always, by definition, follow self evident truths.
But as the universe follows the truths of turtles and humans, we should at least
recognize these truths say nothing about the universe and everything they do
say is reflection of humans (or turtles).
 
  • #62
Pythagorean
Gold Member
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Ok, since we've been getting personal, we should separate our personal arguments from the factual arguments. Anytime you question somebodies motives you're always going to be at least partially wrong (not finger-pointing here, it's a natural result of debate when people have the genuine interest that you spoke of).

So we'll retain your format and do personal first, then topic. I don't think the personal is irrelevant to the topic, but it would be nice to keep them separate.

Now, hopefully you're clear that "One math works in one frame, the other math works in another frame" is an argument against attitudes like this:

George Powell said:
Our universe follows rules, these rules (as far as we can see) have always been obeyed. All of these rules so far can be defined through mathematics, so I presume that the fundamental rules of the universe are 'written' in a language similar to maths.
I'm not making an argument about whether or not math is compatible. The argument is that we could have easily designed it to be compatible with our descriptions of reality (which wouldn't be a stretch, considering math arose from studying reality). I also stead in the previous thread we had this discussion that a TOE might convince me (as long as it's not a makeshift pack of algorithms, utilizing math as a tool bag of random tools that you can get a lock open with) that the universe is fundamentally mathematical, which would also show that mathematics is discovered.

It isn't my imagination or some sort of bias that makes me think that you and turbo and others say things like this. You actually say such things.
non-sequitur. I never claimed that I don't "say things like this". We're talking about your interpretation of my the words. By further interpreting them the way you have been, you don't really make the point that you aren't.
Okay, so we're again back to a complaint about the nature of mathematics that is based upon its utility to physicists and difficulties that physicists encounter when trying to apply mathematics.

We've now come full circle back to the first thing I was saying in this thread, "But physicists erroneously applying mathematics to a problem of science isn't the same thing as a flaw or limitation in mathematics itself" which you [post=1980511]quoted and asserted was bias[/post].
You see, the bias is that physicist's are "erroneously applying mathematics". That's assuming your conclusion, since mathematics would have to be necessarily discovered, and not invented for there to be a wrong way to apply mathematics.

The way I use mathematics is like a sculpture, cutting my function to fit reality. I chip away from functions with other functions and approximate. There's no "erroneous application" since what I do works to the degree I need it to. If I need more accuracy, I cut more and approximate to a higher order.

The claims I make about the sort of things that physicists say, which you keep claiming is some sort of biased view or assumption on my part, you actually keep repeatedly saying.
Or... you keep repeatedly interpreting with a bias...? We could argue circles all day about this, really.


I understand that you were mistaken in what you thought I was saying but that doesn't mean that I'm somehow biased - I am describing the way that you and (some) other physicists do actually behave. When it serves your purpose, in discussing the fundamental nature of mathematics you will very readily blur the line between mathematics and applied mathematics within physics, and imply that some problem that physicists run into in applying mathematics extends to being a general problem with mathematics itself.
But you were interpreting my posts as absolutes, such as "math does not work" when I was the argument really has to do more with are humans the ones making mathematics work?. You were taking my arguments to be me extreme than I intended them.

Bah, I've been responding to the wrong post. I'm going to try do this competently, later.
 
  • #63
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Now, hopefully you're clear that "One math works in one frame, the other math works in another frame" is an argument against attitudes like this:

Our universe follows rules, these rules (as far as we can see) have always been obeyed. All of these rules so far can be defined through mathematics, so I presume that the fundamental rules of the universe are 'written' in a language similar to maths.
What do you mean exactly by ""One math works in one frame, the other math works in another frame"
 
  • #64
Pythagorean
Gold Member
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I'm just going to continue this as if it's part of my last post. But my views have changed a bit since even then, or at last become more clear to me.

When physicists tried to apply the mathematics they used in classical physics at the quantum level, that was an erroneous application of mathematics. But despite the context in which you brought it up, it has no bearing on whether or not mathematics is discovered nor invented. Nor even were someone proposing that there's some matrix-like mathematical substrate to the universe, would it have anything to do with that.
These are your words, not mine. I have no idea what that means. But If you'll look at the author of this thread, and where I quoted him in my last post, maybe you'll see where you're wrong if you consider what I'm actually saying (in fact, look at the first sentence in this whole thread) and not how you took it in and spat it back out (unless that's actually what you meant... compared to GeorgePowell's statement). I honestly thought you were defending GP's views.

However, I still hold that the axioms of mathematics are invented. I cannot deny that there's lots of discovery involved in pi and e, but constant values hardly represent all of mathematics, so it's not basis for "mathematics is discovered" it's basis for "constants are discovered".

So the argument "mathematics is invented" is equally invalid, since you can't bullpen mathematics into a simple little category; There's a lot of elements to mathematics and I find it hard to believe that you and I share the same set of elements everytime we hear the word mathematics. My focus is on the axioms.

I've been looking to counter two arguments of yours:
"mathematics is not invented"
"mathematics is discovered"

Which I still hold are false statements. However, I also realize the approach I've been using is to prove two things by counter-example:
"mathematics is not discovered"
"mathematics is invented"

which is equally false because we're both assuming that mathematics is something we can just lump into a generalization like that.

I will reply to your organized argument eventually if it's necessary, but I still have things to think about for it.

GeorgePowell said:
What do you mean exactly by ""One math works in one frame, the other math works in another frame"
by "works" I mean 'is useful in making predictions in physics'

by "one math" and an "other math" I mean this equation or that equation. This model or that model. There's no ultimate equation that's fundamental to all reality. There's several ways you can go about doing it.

Mathematics is a language. You can use it to describe anything you want, only quantitatively instead of qualitatively. You have some creative license with how you mathematically describe physical things, so long as the predictions work within a given confidence and accuracy.

Language, qualitatively, is the same way: you can't say a fish is a boot. Some people may argue that you can, but if they refine the accuracy of their argument, they might say "a fish-shape can be made out of a boot" or "a boot can be made of a fish". But we all know that a fish is not a boot, so there's still false and true statements and accuracy associated with qualitative description.
 
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  • #65
Pythagorean
Gold Member
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Premise 1: As you yourself affirm, we agree that any intelligence, even an intelligence entirely unlike that embodied by the human brain, which could formulate and understand a particular set of mathematical axioms would arrive at the same conclusions humans arrive at based upon those axioms.

If Premise 1 is true independent of the point in history such a hypothetical scenario might occur at then in the absence of humankind, before its existence or after it perishes, the same alien intelligences would arrive at the same conclusions based upon the same axioms.

Ergo, the relationship between the set of mathematical axioms and the conclusion is a constraint that is independent of the existence of humans or human brains or human-like brains.

If something is independent of the existence of humans it would not be characterized as invented by humans; hence it would fall within the opposite of "invented", which by the conventions of this discussion has been designated "discovered".

the relationship between the set of mathematical axioms and the conclusion is a constraint that is discovered rather than invented.​
I saw this movie (can't remember the name) where they make contact with aliens. Their greeting ship was fixed with lights that lit up in some "mathematical pattern". The point was made in dialogue that mathematics would be the way we'd have to communicate with aliens because it transcends language. I think this is a silly idea. Rote will always be the way to communicate with someone who's language you don't have a codec for. That's how the codec's (standard word by word translations) came about. I just wanted to put that out there. I know you haven't made the argument or anything, but you could have something to say about that.

p1 is a weak premise. The argument for invention is that mathematics comes from consciousness. It doesn't really matter which consciousness it comes from, but it's inherent to one of the ways we think about the world (specifically in series and linearly). The mathematics that tell the most accurate truth about the physical universe are the ones that say the least... that is they have infinite solutions... for then you can chose the one that fits your situation and throw away the others.

p2 is a case of p1

In your "Ergo", you've softened your stance to a point by talking about the relationship between the subject we were debating and "conclusions", then following into your next paragraph you've actually bravely avoided declaring that "mathematics is discovered"

your first post, back in our old thread:
Well, I'm inclined to try staking out the position that it's entirely discovered.
https://www.physicsforums.com/showpost.php?p=1610409&postcount=14

you even put the last two words in red. In fact you're wording is very similar to mine, you say "inclined", but then emphasize the point. It appears you have been doing what I noticed I was doing. Taking the extreme side out of ignorance. You slowly gave up parts of it, and I gave up parts of my stance.

And note that again this does not depend on any aliens actually existing. I am demonstrating that based upon a statement that you agreed with, in the way you yourself view mathematics some part of it has existence independent from humans.
I can agree to that. You have moved toward the middle as much as I have in this debate. We're at least not stating the absolutes anymore.
 
  • #66
This goes way back too the math, and reality thing again. Putting them together. What are everyones views on Heisenburg's principle that focuses on realation's between quantities which in principle are observable thus having formulation of quantum mechanics., the difference between 2piextimesp and ptimes2piex would be imaginary. right?
 

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