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

[Chrisc]

Ditto

Phrak, please feel free to correct me as I am finding it a little scary that I understand such hyperbole.

The notion of numbers and what they represent must exist exclusive of humans if you hold to the idea that mathematics can define the universe before human existence.

But as numbers are not physical, all 4s are indistinguishable.
Therefore 4 represents anything and therefore nothing.

In other words, until humans attach significance to any 4, any equation it takes part in has no
meaning at all.
So to answer your OP, no logic, no truth and no reason exists before we discover them because
discovery in this sense is creation, the creation of reason.

 

[Georgepowell]

Phrak, please feel free to correct me as I am finding it a little scary that I understand such hyperbole.

The notion of numbers and what they represent must exist exclusive of humans if you hold to the idea that mathematics can define the universe before human existence.

But as numbers are not physical, all 4s are indistinguishable.
Therefore 4 represents anything and therefore nothing.

In other words, until humans attach significance to any 4, any equation it takes part in has no
meaning at all.
So to answer your OP, no logic, no truth and no reason exists before we discover them because
discovery in this sense is creation, the creation of reason.

Discovering a rule does not mean that the rule was not obeyed before we discovered it. So that rule has already existed. The mathematics that we use to describe that rule is only our interpretation of it, written in our own language of logic, and does not define it. (This might be slightly different to my OP, but my opinion has shifted slightly...)

The system/language that does define the 'rule' distinguishes between different types of 4, because as you said, 4 can have an infinite number of interpretations.

The 4 that is used in describing the pattern in our brain that creates happiness, is the same 4 that describes the width of an object. Untill we as humans mention the difference.
---------------
To summarise my changed idea (in a more logical order):

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.

I don't know if we agree now or not... But reading our two posts it actually looks like I am on your side now.

Any more criticisms from anyone?

 

[CaptainQuasar]

yes, I do have to restate that, "pure mathematics is void of the human experience of reality". Nobody can make claims about the reality we don't experience.

I think that this is still assuming your conclusions, even if you tack on that "human experience" qualifier.

If mathematics has nothing to do with some fundamental reality, but it also does not arise from a human perspective or experience of anything, where the heck is it coming from? Surely at least our formulation of geometry has something to do with us experiencing flat three-dimensional space through our senses.

That's my fundamental problem with the statement "mathematics exists independent of human thought". There's no way to know that. Note: it's just as ridiculous to make the claim that mathematics is completely dependent on human thought, but I'm not saying that. I'm showing it is dependent on human thought (barring the "completely" which I'm not able to comment on).

If you were to refer back to oldman's [thread=215118]The Question : is mathematics discovered or invented?[/thread] thread where we first discussed this (sorry I never replied to your last post in that thread btw), you would see that I don't insist that mathematics is in its entirety independent of human thought either. It's not like I'm proposing that mathematics texts as a group are some infallible transcendent revelation the way Islam regards the Koran or a Christian Biblical fundamentalist regards the Bible.

Of course the human formulation of mathematics is going to be thoroughly human in its nature, colored and maybe even twisted by the human perspective on the universe, and undoubtedly encompassing only a portion - perhaps a tiny portion - of its subject matter. As I said [post=1618613]here[/post] the point is that the subject that human mathematics studies is something that has independent existence. It's not studying something that is a human creation like French Renaissance Literature is studying or scholars of Early 20th Century Film are studying.

But I didn't say that, and I've done my best to make the point against that. The flaws are intrinsic to science.

You haven't said so in this thread, but turbo said that bit I quoted above and in the other thread [post=1649585]you made[/post] what looks to me like an exactly parallel argument: you told me that "what you see as mathematics is a consequence of your brain having developed in the macroscopic world" and similarly to turbo above talked as though mathematics stops working at the quantum level, which as I've pointed out repeatedly it does not at all.

So I understand just fine if you're recanting that now but this is what I've been talking about: both you and turbo acted as though, since the development of understanding of quantum level phenomena was an Earth-shattering paradigm shift for physics and chemistry, it's had a similar impact on mathematics or as if the behavior of quantum-level phenomena invalidates some aspect of mathematics.

But that's not true at all; as I've emphasized before, not one iota of mathematics had to be scratched out and rewritten in the face of quantum phenomena the way that so much of physics had to be. This is why statements like the ones I've referenced look like psychological projection on physicists' part. Yes, quantum phenomena are incredible and rock-your-world type things but they don't have bearing on the nature of mathematics.

I guess hidden in here, I'm defending science where you have assaulted it.

I would say that if anything I have assaulted scientists if anything, not science itself. I think it's an entirely forgivable and understandable transference to have made. I only point it out because we're talking about the nature of mathematics, not the nature of physics, and the way that quantum phenomena forced such a revolution in the understanding of physics just doesn't have bearing on the nature of mathematics nor does it present any evidence regarding whether or not the things that mathematics studies have existence independent of human thought.

It's true that I've been championing the other side in an effort to bring you to the middle. I should be focusing on showing you how you're wrong...

Do you think maybe that the reason you're having so much trouble convincing me, and why you have to do things like apparently recant the quantum phenomena argument, is because I'm not wrong? That perhaps the things I'm saying are true and so would have to be incorporated into or accounted for in whatever this "middle" position is, which you've evidently been secretly holding all this time?

Then we agree somewhat. But I still see no connection between reality and pure mathematics. Everytime I say this, it's an opportunity for you to show me the connection.

I did quite a bit of showing you the connection in that previous thread, during which you mostly said that you were still formulating your thoughts on the subject. But it's an interesting topic so it's well worth re-hashing.

Okay - so, even if there were no humans around to see it, the universal gravitational constant averaged across all gravitational interactions in the universe would still approach

, right? Obviously the expression above is a human formulation of that value, but there is some constant ratio between that value and the averaging value of, say, all the weak nuclear force interactions in the universe, right?

Even if there were some group of aliens who had an innate understanding of GR spacetime geometry and to whom the concept of "gravity" never even arose, the universal gravitational constant we speak of and the way we use it would not contradict their understanding of physics - at worst it might appear as a silly and pointlessly arbitrary abstraction of marginal importance to them but it would be consistent with their knowledge of the way the universe works. So this is a scientific fact that I think we can say exists independent of humanity. At least it's more independent of humanity than some fact about French Literature or Early 20th Century Film.

In a nutshell, what I'm saying is that π, as a mathematical fact that is the limit to which the averaging ratio of the radius to the circumference of all circular objects in the universe approaches, is as independently real and existent as is G. And that there are a whole network of relationships underlying the human formulation of mathematics that are equally as independently real and existent.

Note that the π ratio being a fact from pure mathematics, from not only geometry but trigonometry as well, that has relevance in reality as experienced by humans, this is a counterargument to your "void of... reality" claim above.

I had all sorts of elaborate ways of expressing and refining the idea that I used in the other thread but I'll leave those aside until you've responded to the above.

I absolutely disagree. The most broad principals in religion are:

1) there's an omnipotent entity (their used to be several, but that axiom must have led to more inconsistencies within the system of religion somehow and was eventually rejected).
2) there's an afterlife (reincarnation included... the fundamental concept being you don't die when you "die".)
3) there's an objective moral basis

By doing the right 3), 1) allows you into a 2) that you'll like better. If you don't obey 3), 1) decides you will have a remarkably uncomfortable 2).

To me this looks like a crude attempt to paper over the extreme differences in human religion in an attempt to pound the round peg of religion into a square hole so that it has the similarities to mathematics that you want to claim it has.

1) simply is not universal - you're talking like someone from a Judeo-Christian religion. The Dali Lama talks about all sorts of different gods when he discusses Buddhist theology. But Islam says "There is no god but Allah."

Some Buddhists and Hindus and other polytheist say "One supreme god up above all the others? Uh... sure! That's kinda like bráhman." And I'm sure many of them have incorporated a single supreme being into their theology since exposure to Judeo-Christian religions. But the theological definition of bráhman is nothing like the omnipotent supreme being you are positing as common to all religions there.

And of course, going back in history there are many variations. In early Zoroastrianism, for example, there were two equally powerful beings, one that was pure good and one that was pure evil in modern terminology.

Even within Judeo-Christianity there's lots of conflicting theology. Jews, Trinitarian Christians, and Mormons all consider themselves to be monotheistic but Jews consider Trinitarian Christians to be polytheistic and they both consider Mormonism to be polytheistic.

I will grant you that 2) is pretty common but I would not say that it's universal. I have been told by some modern Jews that "For all we know, there is no afterlife. This life is a reward itself - that is the covenant with God."

Your 3) is definitely another attempt at mushing things from different religions together. Modern and Christian-influenced notions of sin and redemption and absolute good and evil just aren't the same thing as the notions of making sacrifices to please the gods in so many religions. Nor is it the same thing as the early Greek or Viking ideas that winning glory in battle might get you picked out as a sort of trophy by the gods to decorate their heavenly abodes.

And a way to win a good afterlife isn't universal either - some Greeks thought of Elysium as I mentioned there, but in the Odyssey, through sorcery Odysseus meets and speaks with the shade of Achilles in Hades. Achilles won more glory than any other figure in Greek mythology - but he was confined to the grey netherworld of Hades with everyone else, thirsting after the life he once had. Some scholars think that the afterworld in early Judaism, sheol, was like the Greek Hades - simply the place that dead people went, nothing good about it at all.

So yeah, I personally think that many if not most modern ideas of multicultural equivalence are politically correct reductionism that do not hold up under close examination. Yeah, there are some things that all people have in common, but definitely all religions are not equivalent and interchangeable at some level.

What you consider mathematics may be the tip of the iceberg of something much more fundamental to reality; what you consider the fundamental axioms of all mathematics could be laughable to an alien species as a skewed view of a special case, because you chose the axioms that were attractive to you, as a human, not seeing the most general case of the axiom.

The alien race may very well have some sort of dynamic axiom system, in which mathematics is one of a thousands of stable states in the system.

I've never said anything about there being some fundamental true axioms of mathematics. Nor that human understanding of mathematics is in any way comprehensive. So don't worry, you're not surrendering anything at all, if indeed you've really had the same secret position the whole time and your most recent statements aren't the result of me presenting evidence incompatible with your earlier views.

You really need to go back and re-read that other thread. The way in which I have proposed that mathematics has an independent existence is pretty abstract.

⚛​

 

[Pythagorean]

Well, I had a reply written out, but I somehow lost it in the log-in process, so I've lost motivation now. I will reply in more detail later, but I wanted to clear up something that I think is important to how you're interpreting my posts.

I've never said anything about there being some fundamental true axioms of mathematics. Nor that human understanding of mathematics is in any way comprehensive. So don't worry, you're not surrendering anything at all, if indeed you've really had the same secret position the whole time and your most recent statements aren't the result of me presenting evidence incompatible with your earlier views.

When I said my thoughts were undeveloped in the previous thread, perhaps I didn't make it clear that the whole point of my engagement in this discussion is to develop my thoughts, so of course my views change as you present logical arguments (and no evidence has been presented in my opinion, this is a philosophical debate, its comes down to logical arguments, not "evidence" in the scientific regard.)

If you're thoughts haven't changed during the course of the discussion, then you're still holding firmly to your bias (which is what I was pointing out). Just because I don't stop and validate your argument doesn't mean I still disagree with them. I'll usually just drop an argument if I see it's flaw and continue on with the rest. If I state that I agree it's usually because I don't see how it follows (i.e. your point was true, but not valid).

My thoughts have definitely moved more towards the middle through discussion with you (on a side note, nobody else has really contributed in this regard so there's a chance I"m just believe your flawed arguments because you've been persuasive). I will respond in detail to your finer points later. I'm intellectually exhausted from the reply I misplaced.

I also tend to play Devil's Advocate where I'm ignorant on a subject. This prompts more valuable (to me) discussion as replies tend to cut straight to the point.

I'm a conventional troll (as opposed to the modern trolls who just seek to insult and aggravate without any sort of interest in knowledge).

 

[CaptainQuasar]

Well, I had a reply written out, but I somehow lost it in the log-in process, so I've lost motivation now.

Oh, man, I hate it so much when that happens. My condolences.

But from what you said there, I have to maintain that your declarations that I am biased or that my arguments are generally flawed appear to simply be rhetorical with nothing to support them.

I mean, if not only am I able to point out specific flaws in your arguments, but I can now anticipate that your arguments and position are going to be changing over time, what reason at all do I have to be swayed by them? How is it being biased to not be swayed by that?

Anyways, I do hope you get the opportunity to re-write your response, but I know exactly what you mean about being exhausted and I'd understand if you didn't.

P.S. I do not think you're a troll, even though you've exhibited some contrarian behaviors (which I exhibit myself as well.) The thing about a troll is that a troll is contrary solely for the purpose of getting attention and keeping the argument going. I believe that you are earnestly interested in exploring the nature of mathematics.

P.P.S. My thoughts have changed during the course of the argument but it's that I have refined them in the course of articulating them.

⚛​

 

[Pythagorean]

Oh, man, I hate it so much when that happens. My condolences.

But from what you said there, I have to maintain that your declarations that I am biased or that my arguments are generally flawed appear to simply be rhetorical with nothing to support them.

I mean, if not only am I able to point out specific flaws in your arguments, but I can now anticipate that your arguments and position are going to be changing over time, what reason at all do I have to be swayed by them? How is it being biased to not be swayed by that?

I don't think your arguments are generally flawed; If I recall correctly, I was being picky about certain parts of your arguments. I'm not a mathematician, and the definition of mathematics is squirmy to me. I've never taken a proofs class. I've utilized plenty of math in many different forms, but it's very much a rugged tool to me, and like a lego set it can be manipulated into the shapes you need.

A sound argument is a sound argument no matter who it comes from or what their history with you is. There's no guarantee that the flaws you point are completely founded, either.

I also am not making an argument for your bias yet. I stated it as an intuitive guess. Another aspect of my debate technique is to make intuitive guesses so they can be analyzed later when argument has developed.

Also, you may be a better arguer than me (I suspect this is true), which itself does not validate your claims.

Lastly, I've noticed sometimes you'll assume comments I make are in support of the antithesis when they're not. I'm testing your claim, but I don't really have an alternative.

The thing about a troll is that a troll is contrary solely for the purpose of getting attention and keeping the argument going. I believe that you are earnestly interested in exploring the nature of mathematics.

Yes, I am interested in the nature of mathematics, but I'm also kind of blind on the topic so I feel like a troll when I'm fishing around in the dark for an argument.

(back to previous post...)

If mathematics has nothing to do with some fundamental reality, but it also does not arise from a human perspective or experience of anything, where the heck is it coming from? Surely at least our formulation of geometry has something to do with us experiencing flat three-dimensional space through our senses.

I did say "pure mathematics"

mathematics (minus the pure) arises the same way language arises. You want to learn more about an observation so you study it it, and you find relationships. At first, you may use words like "that's a lot, that's a little, that's none", but then as you become more accurate about your observations, you can use a numerical system.

But it also arises the same way rules in a board game arise. As long as you're careful enough when you define the system of rules, they won't contradict themselves. This doesn't mean the rules of the game were discovered.

"Pure mathematics" is the further exploring the implications of those other rules to make more rules. Just because we can use the rules to better articulate our observations doesn't mean that they inherently came from our observations.

As I said here the point is that the subject that human mathematics studies is something that has independent existence. It's not studying something that is a human creation like French Renaissance Literature is studying or scholars of Early 20th Century Film are studying.

ok, give me more.. what is that "subject" that mathematics is studying? I don't see the ghost, so I can only suspect that you're imagining things that aren't there.

you told me that "what you see as mathematics is a consequence of your brain having developed in the macroscopic world" and similarly to turbo above talked as though mathematics stops working at the quantum level, which as I've pointed out repeatedly it does not at all.

This argument: "what you see as mathematics is a consequence of your brain having developed in the macroscopic world" I can rearticuilate. The only real burden their is "macroscopic world".

I used plenty of math in QM, that's not my argument. My argument before was that the world is non-determinant (I had thought QM said this too, but you were right, it does not unseat determinism... which makes sense since ). If you want to have a discussion (heh,d ebate!) about how non-determinism implies mathematics is not discovered, we should start a new thread.

Anyway, I think what me and turbo are both trying to do is preemtively dismantle the argument that some mathematicians make (you're not the only one I've had this debate with... oddly it's always mathematicians defending this claim).

That argument suggests that mathematics describes all of reality on some universal level or that mathematics is somehow reality. 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. The math reduces inaccuracy and uncertainty, but it does not eliminate it. Technically, this has nothing to do with our argument, but it's a common misconception of science-dumb mathematicians that can be the justification for their claim that math is discovered, not invented. (and yes, I realize that I'm a math-dumb scientist).

Ok, have some homework to do (applied analysis, no less!) so I'll do more later.

 

[CaptainQuasar]

I don't think your arguments are generally flawed; If I recall correctly, I was being picky about certain parts of your arguments. I'm not a mathematician, and the definition of mathematics is squirmy to me. I've never taken a proofs class. I've utilized plenty of math in many different forms, but it's very much a rugged tool to me, and like a lego set it can be manipulated into the shapes you need.

A sound argument is a sound argument no matter who it comes from or what their history with you is. There's no guarantee that the flaws you point are completely founded, either.

I also am not making an argument for your bias yet. I stated it as an intuitive guess. Another aspect of my debate technique is to make intuitive guesses so they can be analyzed later when argument has developed.

I think it's entirely fair to make intuitive guesses or even to mention them in an argument. It's just that when you broadly declare that I'm biased, without demonstrating how my bias is prejudicing my judgment or arguments, nor even specifically saying what you think I'm biased about, it certainly appears like a rhetorical technique and of course I'm going to respond.

(Technically it would be an ad hominem fallacy: it looks as though you're implying that there's something wrong with my arguments because there's something wrong with me, that I have an inherent bias which must be producing un-identified flaws in my arguments.

As an aside, I find that most people use the phrase ad hominem incorrectly, as if it's something to do with etiquette, that any criticism of another person or their behavior is somehow erroneous or wrong. But that has nothing to do with it. I think it's perfectly valid for someone else to advance personal criticisms of me, or vice versa, particularly if there's evidence to cite.)

Also, you may be a better arguer than me (I suspect this is true), which itself does not validate your claims.

Definitely not, you're right. An argument must stand on its own and is valid exclusively through its own virtues.

As far as me being a better arguer that you, I ought to mention that in picking theology as an analogy to mathematics you had the bad luck to hit on something that is sort of a hobby of mine, I have spent a great deal of time studying the various religions of the world.

I did say "pure mathematics"

mathematics (minus the pure) arises the same way language arises. You want to learn more about an observation so you study it it, and you find relationships. At first, you may use words like "that's a lot, that's a little, that's none", but then as you become more accurate about your observations, you can use a numerical system.

But it also arises the same way rules in a board game arise. As long as you're careful enough when you define the system of rules, they won't contradict themselves. This doesn't mean the rules of the game were discovered.

Hmmm. The thing is, it seems that people in widely disparate civilizations around the world have been able to consistently and repeatedly come up with sets of rules, independently, that do not contradict each other. In the case of mathematics it does not seem to require anywhere near as much care or caution to avoid coming up with contradictory rules.

For example, when you talk to people about the way that they perform simple arithmetic, you find that there are a widely varying array of algorithms that are used. But they all arrive at the same answer. And if someone were to arrive at a different answer, it's possible to objectively prove that the answer is incorrect, without needing to know the details of the algorithm used to derive it. (Though of course you might look at that algorithm if you're teaching someone arithmetic and trying to help them find where they're going wrong.)

"Pure mathematics" is the further exploring the implications of those other rules to make more rules. Just because we can use the rules to better articulate our observations doesn't mean that they inherently came from our observations.

I guess we're working with somewhat different definitions here: I would pretty much regard pure mathematics to be anything that is beyond the realm of applied mathematics. Simply, say, learning about prime numbers and looking at how they relate to factoring in the case of multiplication, for example, when you're sort of examining how the math works instead of using arithmetic to count physical things or measure physical quantities, seems like pure mathematics to me.

In any case, to get back to my geometry example - if, say, the Pythagorean theorem as something derived from the implications of other rules of Euclidean geometry qualifies as pure mathematics, the fact that it also works properly in the physical world seems to me to violate your assertion that pure mathematics is void of reality.

ok, give me more.. what is that "subject" that mathematics is studying? I don't see the ghost, so I can only suspect that you're imagining things that aren't there.

Wait, so are you saying that you don't believe there is a subject that mathematics is studying? When a mathematician is examining a problem and trying to find a proof or other solution, what is she examining? That's what I would say that mathematics is studying, the thing or things that mathematicians examine all the time and try to formulate rules about and descriptions of.

This argument: "what you see as mathematics is a consequence of your brain having developed in the macroscopic world" I can rearticuilate. The only real burden their is "macroscopic world".

I used plenty of math in QM, that's not my argument. My argument before was that the world is non-determinant (I had thought QM said this too, but you were right, it does not unseat determinism... which makes sense since ). If you want to have a discussion (heh,d ebate!) about how non-determinism implies mathematics is not discovered, we should start a new thread.

Okay... but can we at least agree that the discovery of QM did not result in any part of mathematics being invalidated, and that the effect, as far as invalidating anything, was entirely within physics?

Anyway, I think what me and turbo are both trying to do is preemtively dismantle the argument that some mathematicians make (you're not the only one I've had this debate with... oddly it's always mathematicians defending this claim).

I'm not a mathematician, btw, I'm a software engineer. Though my degree in school was a combination of pure mathematics and computer science.

That argument suggests that mathematics describes all of reality on some universal level or that mathematics is somehow reality.

Sort of along the lines of "the universe and the fundamental reality is an information matrix" kind of thing? I'm not advancing that, as I said I don't think that the existence of physics phenomena that are impossible to model or describe with mathematics would invalidate the sort of things I'm saying. I'm really just saying something along the lines of π and G are both real and existent independent of humanity (though not that they necessarily have the same nature.) I'll declare right now, that by no means entails that the fundamental nature of the universe is mathematical.

(I certainly find the notion that the universe is entirely mathematical at base tempting, but I don't think that would be anywhere near as demonstrable or supportable as what I'm saying and so that's not what I'm arguing in this context.)

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.

I don't think that's true of everything, is it? I thought that wave mechanics, for example, is equally applicable at both the macroscopic and quantum levels. And of course simple things like π work and appear in radial equations all over the place. And I know that there are massive differences in classical field theory versus quantum field theory, but I believe they're using as tools many of the same mathematical elements - scalar, vector, tensor, and spinor fields, of course, at the very least.

(and yes, I realize that I'm a math-dumb scientist).

I wouldn't be surprised, since I've been out of school for a few years and the opportunity to use any really good math stuff in my work is pretty infrequent, that you probably have a much better handle on many mathematical topics than I do right now.

⚛​

 

[Pythagorean]

(I've made two posts here, just fyi before you hit the quote button)

So I asked my math teacher today, and of course his view is that math is discovered. He drew on the analogy of language (which I consider math... a highly articulate language) before I even brought that metaphor up.

His argument was that even people of different languages have an objective basis and will eventually be able to correlate the languages (otherwise communication between different language-speaking peoples would be impossible).

This is aligned with your argument:

Wait, so are you saying that you don't believe there is a subject that mathematics is studying? When a mathematician is examining a problem and trying to find a proof or other solution, what is she examining? That's what I would say that mathematics is studying, the thing or things that mathematicians examine all the time and try to formulate rules about and descriptions of.

But to me, what you're effectively saying is that languages (math included) are used to describe something that is objectively real (i.e. reality). This does not seem to satisfy the claim that "math is discovered" for me. It seems to say instead, what you explicitly stated before:

As I said here the point is that the subject that human mathematics studies is something that has independent existence.

That sounds to me, like you're admitting that math itself is not discovered (which is what I thought we were debating). I feel that it is an abstraction that was invented to study that underlying thing.

The Physicist's Analogy:

I'll start with a quote from you:

I wouldn't be surprised, since I've been out of school for a few years and the opportunity to use any really good math stuff in my work is pretty infrequent, that you probably have a much better handle on many mathematical topics than I do right now.

I guess what I meant in my post to which this reply originated is that I'm math-dumb about the fundamental axioms of mathematics. I've never studies the formal foundation. It's been exclusively used as a tool in my experiences.

Now, tools are real, but they didn't exist before humans. In some cases, they can just be an abstaction of a real object. For instance, a shillelagh is a tool, but the real object is just a stick. Not much is done to make a stick into a shillelagh physically, but there's no such thing as a shillelagh without a human (or a primate in some cases) to see the stick as a weapon that can be used to hit things.

A spear is just a couple shavings away from a stick, but there's a physical interaction involved here.

Being physically trained, I've always viewed mathematics as a tool. Now, that tool may be made of things that were around before the human, but the tool itself was constructed (i.e. invented) out of those previously-existing substances.

Sort of along the lines of "the universe and the fundamental reality is an information matrix" kind of thing? I'm not advancing that

Well that kills a few arguments of mine then. I apologize for operating on this assumption, and also for not making the assumption clear before so that you could refute it.

 

[Pythagorean]

I'm really just saying something along the lines of π and G are both real and existent independent of humanity (though not that they necessarily have the same nature.) I'll declare right now, that by no means entails that the fundamental nature of the universe is mathematical.

I'm pretty sure there's no such thing as a straight line or a perfect circle anywhere in the universe. Space is also noneuclidian (G is based off of euclidian space). I realize pi has other applications, but these are just examples of where those constant work as a tool, but don't perfectly describe the reality.

(warning: intuition involved above!)

I don't think that's true of everything, is it? I thought that wave mechanics, for example, is equally applicable at both the macroscopic and quantum levels. And of course simple things like π work and appear in radial equations all over the place. And I know that there are massive differences in classical field theory versus quantum field theory, but I believe they're using as tools many of the same mathematical elements - scalar, vector, tensor, and spinor fields, of course, at the very least.

That example doesn't follow from what I'm talking about. Looking at a wave alone can be helpful in the specifics of a single, isolated wave (which doesn't exist in nature... or at least there's no way we could observe it if it did...). For instance, you can use it to model a guitar string, but if you try to model several guitar string in reality, you have to change the maths to get a decent prediction out of the whole system. In music, they have to use techniques like http://en.wikipedia.org/wiki/Equal_temperament" to get the instrument to act (disguise itself) like the beautiful math version. We actually can't hit the notes like we'd be able to in a math world, but the human ear is insensitive enough to be tricked by a system that is just slightly off the ideal.

This is what I mean when I say math doesn't have anything to do with reality. That statement is a bit extreme. What I really mean to say is that it takes creativity and "leaps of faith" to bridge the gap between reality and math.

In fact I remember a neurological argument made by Penrose (I think) that pure logic would have failed in making any predictions in science. It requires a leap of faith in the human brain for science to be useful at all.

 

[CaptainQuasar]

So I asked my math teacher today, and of course his view is that math is discovered. He drew on the analogy of language (which I consider math... a highly articulate language) before I even brought that metaphor up.

His argument was that even people of different languages have an objective basis and will eventually be able to correlate the languages (otherwise communication between different language-speaking peoples would be impossible).

This is not anything I'm saying. I would not regard language to be the same as mathematics any more than I would regard physics to be the same as mathematics. (I'd say that they're less similar, in fact.)

But to me, what you're effectively saying is that languages (math included) are used to describe something that is objectively real (i.e. reality). This does not seem to satisfy the claim that "math is discovered" for me.

*snip*...

That sounds to me, like you're admitting that math itself is not discovered (which is what I thought we were debating). I feel that it is an abstraction that was invented to study that underlying thing.

This is again something that got discussed extensively in the other thread. Of course all the words and symbols and things that humans use to express mathematics are invented. It's not the words and symbols and things, nor even the general human formulation of mathematics, that I am proposing as independent of humanity.

Physics is used to describe something that is objectively real; does that mean that physics is invented?

If you group physics, mathematics, and Swedish grammar together by the degree to which each examines something that exists independent of humanity, it's physics and mathematics that go together, not mathematics and Swedish grammar. As I said in the other thread this is what I believe the "discovered or invented" question is asking.

By bringing in things that your math professor says, or that you've heard other mathematicians say, and arguing against those things it seems like you're trying to avoid understanding or dealing with what I'm saying, which is different.

Now, tools are real, but they didn't exist before humans. In some cases, they can just be an abstaction of a real object. For instance, a shillelagh is a tool, but the real object is just a stick. Not much is done to make a stick into a shillelagh physically, but there's no such thing as a shillelagh without a human (or a primate in some cases) to see the stick as a weapon that can be used to hit things.

If a human picks up a rock and uses it as a doorstop or to pound a nail into a piece of wood, that tool existed before any humans existed. Humans using something as a tool does not change its fundamental nature, just as I've been saying that physicists using mathematics as a tool does not change the nature of mathematics.

And even besides that, this line of reasoning, like the "human mathematics is used to describe something" one above, does nothing to distinguish physics from mathematics. I could just as easily say that physics is a tool that mathematicians use when they want to apply mathematics to the more experiential parts of reality, hence physics is just a tool and it's invented.

I'm pretty sure there's no such thing as a straight line or a perfect circle anywhere in the universe. Space is also noneuclidian (G is based off of euclidian space). I realize pi has other applications, but these are just examples of where those constant work as a tool, but don't perfectly describe the reality.

See, you're again ignoring most of what I've already said. (Though I've said quite a bit, so it's understandable.) This is why up above I used the big clumsy phrase where I expressed π as the limit to which the ratio of the circumference to radius of all the circle-like things in the universe approaches as they approach the shape of a perfect circle.

There's no case where the value of G is perfectly exact either. That's why the value for G I listed above has a tolerance range appended to the end of it. You haven't demonstrated anything that disinguishes π as different from G, so I maintain that they are independently real to the same degree.

That example doesn't follow from what I'm talking about. Looking at a wave alone can be helpful in the specifics of a single, isolated wave (which doesn't exist in nature... or at least there's no way we could observe it if it did...). For instance, you can use it to model a guitar string, but if you try to model several guitar string in reality, you have to change the maths to get a decent prediction out of the whole system. In music, they have to use techniques like http://en.wikipedia.org/wiki/Equal_temperament" to get the instrument to act (disguise itself) like the beautiful math version. We actually can't hit the notes like we'd be able to in a math world, but the human ear is insensitive enough to be tricked by a system that is just slightly off the ideal.

This is what I mean when I say math doesn't have anything to do with reality.

Whoa, whoa, whoa, you're switching topics here. The reason I was talking about wave mechanics wasn't to prove anything about reality, it's because you claimed that the mathematics that must be used between the macroscopic and quantum frames is different. But it's not.

That statement is a bit extreme. What I really mean to say is that it takes creativity and "leaps of faith" to bridge the gap between reality and math.

I still maintain that this is assuming your conclusions. The only reason it requires any more of a leap of faith than anything else is if you assume that math is "void of" or unconnected to reality in the first place.

In fact I remember a neurological argument made by Penrose (I think) that pure logic would have failed in making any predictions in science. It requires a leap of faith in the human brain for science to be useful at all.

Well, if you want to say that mathematics is only as discovered as physics is, I'd accept that, it's pretty much what I'm proposing.

⚛​

 

[Pythagorean]

(note: two posts again... )

This is again something that got discussed extensively in the other thread. Of course all the words and symbols and things that humans use to express mathematics are invented. It's not the words and symbols and things, nor even the general human formulation of mathematics, that I am proposing as independent of humanity.

That's quite different to me. The term "mathematics" is not just the symbols and things that human use to express it. Mathematics is the system of axioms itself.

Physics is used to describe something that is objectively real; does that mean that physics is invented?

Absolutely! Physics is not reality! That's a common misconception! It's a collection of techniques used to model reality.

If you group physics, mathematics, and Swedish grammar together by the degree to which each examines something that exists independent of humanity, it's physics and mathematics that go together, not mathematics and Swedish grammar. As I said in the other thread this is what I believe the "discovered or invented" question is asking.

Once again you're making your argument by using the general terms (physics and mathematics) then being specific about the Swedish grammar (like you did with religion earlier). The general term would be language, not "Swedish grammar" is obviously not required. However, language is required if you want to describe physics. Mathematics alone just won't do it. You have to assign meaning from reality (from your observations) to the mathematics.

By bringing in things that your math professor says, or that you've heard other mathematicians say, and arguing against those things it seems like you're trying to avoid understanding or dealing with what I'm saying, which is different.

We both agreed a sound argument is a sound argument. I'm able to separate your arguments in my mind. One of our problems may have been the definition of mathematics. But I declared in this post, above, that mathematics is the system of axioms that humans (invented/discovered) that we're disputing (I thought) not the symbols.

If a human picks up a rock and uses it as a doorstop or to pound a nail into a piece of wood, that tool existed before any humans existed. Humans using something as a tool does not change its fundamental nature, just as I've been saying that physicists using mathematics as a tool does not change the nature of mathematics.

"Humans using something as a tool does not change its fundamental nature". Of course not! It's fundamental nature is a rock. The human was using it as a tool, ignoring its fundamental nature, giving meaning to it. The doorstop was invented. As were pet rocks. They may be lousy, cheap inventions, but they actually take ingenuity to come up (it may seem trivial to you, but you were raised in an educated society so you were handed plenty of prejudices before you ever started thinking for yourself... I'm not different of course, this is human nature.)

And even besides that, this line of reasoning, like the "human mathematics is used to describe something" one above, does nothing to distinguish physics from mathematics. I could just as easily say that physics is a tool that mathematicians use when they want to apply mathematics to the more experiential parts of reality, hence physics is just a tool and it's invented.

So we agree! I've never once claimed that physics is discovered. To me, a soon-to-be physicist, physics is a tool. It works.

Whoa, whoa, whoa, you're switching topics here. The reason I was talking about wave mechanics wasn't to prove anything about reality, it's because you claimed that the mathematics that must be used between the macroscopic and quantum frames is different. But it's not.

Er, no, I don't think I claimed that this thread. I've taken both semesters of QM since last thread and had time to digest the information (by the way, the wave equations are somewhat similar, but not the same as classically. In QM the operators are observables and the function that's operated on is the wave function. Classically, the observables are in the funciton and we don't consider the hilbert space.)

My argument above was not about macroscopic vs. quantum. It's about generalization vs. specifying. We were arguing two different things there.

I still maintain that this is assuming your conclusions. The only reason it requires any more of a leap of faith than anything else is if you assume that math is "void of" or unconnected to reality in the first place.

Well, if you want to say that mathematics is only as discovered as physics is, I'd accept that, it's pretty much what I'm proposing.

⚛​

It sounds like you want me to play fair or something. It was harsh of me to say "void of" or "unconnected to" reality. Let's scratch that from the record. "Is invented" is sufficient for now. But now we're on the level that they're even, eh? They're both invented!

Notice, I never once (not even in the last thread we discussed) claimed that physics is discovered. Physics is a human interface (and an abstraction) for reality and mathematics is a human interface for abstractions such as physics. Mathematics applies to a lot more things than science, granted, but all these things tend to focus on human inventions too (for instance, sudoko). NOTE, I'm saying this to the benefit of mathematics, I'm not saying the sudoko's couldn't be applied to reality, 100% of this discussion doesn't have to be debate. I realize now, looking back at your posts that you've gotten defensive on points that I wasn't even really debating. I am capable of mixing discussion with debate (as are you, when you're aware that I'm not attempting to tie a point into my argument.)

 

[Pythagorean]

Alien Argument:

By the way, every time I've said mathematics did not exist before humans, I've been absolutely anthrocentric.

I kind of assumed everyone agreed on this, but replace human with sentient life. I mean, "alien" can means space mites that will never develop mathematical axioms because they're completely instinctual (as far as we can tell).

Sentient life includes any alien that has a brain very similar to humans and learns and comprehends things the same way humans do. I wouldn't be surprised at all if they invented/discovered mathematics. I also wouldn't be surprised if they invented/developed a moral basis (reward/punishment), which is what's left of my religious analog (religion being different methods of practicing the perspective that morality as an objective/discovered thing).

But like I said, these aliens would have to have brains remarkably similar to ours for them to develop math in such a way that it can be compared to ours.

A monkey has a remarkably similar brain, but it has no concept of mathematics. It's brain does calculations, but he's obviously not doing mathematics in his head. We know this, because our brain does the same thing (a basket ball player taking a shot). He can build axioms to make the shot without mathematics/physics, based on his interactions with reality. His interactions on reality are all that is discovered. His axioms are invented in the most consistent way he can afford to understand those interactions. Those interactions/observations are not fundamentally mathematical in nature just because you can describe them that way (as I think you admitted, but I think that invalidates the alien argument.).

Why should any of the interactions/observations be fundamentally any of the ways we can describe them because a like-minded person comes up with the same idea?

Tool Analogy:

If physics were a chainsaw and the tree were reality, the sharpening tools, and the almighty chainsaw tool (and a plethora of other little tools and your bar oil, and your gasoline) would be mathematics. The chainsaw might go a little while without the tools, but you won't get very far through a single tree (and we're in a forest of a 10^100 trees at least). Mathematics and physics together come closer to pwning reality than either does separately, but I'd suspect things would be a lot less confusing if they actually reached reality.

 

[HallsofIvy]

Mathematical truths exist be for we "find" or prove them but not before we define them.

Is "the sum of the angles in a triangle is 180 degrees" a "mathematical truth"? It is in Euclidean geometry but not in non-Euclidean geometry. There exist an infinite number of "mathematical system" or "axiomatic structures". We create mathematical truth when we define a specific mathematical system. We then discover those truths when we prove them.

Whether or not "a world can exist through purely a system of numbers, values and rules" depends entirely upon what you mean by 'a world'. In any meaning of 'world' that I would use, the answer to your question is "certainly not"!

 

[CaptainQuasar]

Er, no, I don't think I claimed that this thread. I've taken both semesters of QM since last thread and had time to digest the information (by the way, the wave equations are somewhat similar, but not the same as classically. In QM the operators are observables and the function that's operated on is the wave function. Classically, the observables are in the funciton and we don't consider the hilbert space.)

My argument above was not about macroscopic vs. quantum. It's about generalization vs. specifying. We were arguing two different things there.

This is 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.

That's why I was talking about things like wave mechanics and fields, because the math is the same in both frames. It seems to me like you're saying there's a particular kind of math that only works in the macroscopic frame or something like that. I don't think you've demonstrated how math stops working somewhere at some point. Even if you were examining some very tiny circular objects or circular regions of fields, quantum-scale-sized-ones, the ratio between their circumferences and their radii would still approach 2πr.

Perhaps you haven't been trying to say that, but if you haven't then I definitely still don't understand what the discussion of QM in this context is all about.

(And yes, as you sort of pointed out, I know that the "wave function" is a misnomer because what it's describing is not an application of wave mechanics. But it's my understanding that there are other areas of QM where wave mechanics gets applied, though I could be wrong.)

So we agree! I've never once claimed that physics is discovered.

Okay, we had hit on this again and again and again in the other thread, I thought you might remember. If all you're saying is that mathematics is invented and not discovered because anything that describes something else is invented, and that hence anything that is ever communicated between humans is invented, that seems like a pretty mundane meaning of the word "invented" and I do not think that's the meaning of "invented" in the question "Is mathematics discovered or invented?"

I thought that we had already gotten this out of the way about a hundred postings back in the overall discussion, though I guess not. I actually went out of my way to dig up where we discussed this in the previous thread and linked to it above, in the hopes that we wouldn't get mired in it again, but alas...

So of course, this is why I have been talking about the significant thing being whether the subject of mathematics is discovered or invented; I was referring back to that previous discussion of the meaning of "invented". 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.

† General grammar contains contradictions in a fashion that I think is different than anything in human mathematics and is not consistent in the same manner, which is why I would specify "Swedish" the same way I would specify a particular theology. For example, if you know everything about all human mathematics other than Mayan mathematics, you're going to be able to figure out the solution to a math problem expressed in Mayan notation even if you have never seen that problem solved before. You may even be able to correctly determine entire sets of rules that Mayan mathematics would have to contain.

But even if you know every human language other than, say, Basque (let's pick that one since it's a "language isolate" unrelated to other languages in the same way that Mayan mathematical notation is unrelated to other mathematical notations) you are not going to be able to figure out what the rules of grammar of Basque are, nor other conventions of the language like how to pluralize words, unless you have access to examples of that being done in the language.

But like I said, these aliens would have to have brains remarkably similar to ours for them to develop math in such a way that it can be compared to ours.

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.

Now you're again making this sort of statement but I really don't think that you're presenting any sort of evidence or an argument for it; it again seems to me like something which simply assumes your conclusions. 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?

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:

Even if there were some group of aliens who had an innate understanding of GR spacetime geometry and to whom the concept of "gravity" never even arose, the universal gravitational constant we speak of and the way we use it would not contradict their understanding of physics - at worst it might appear as a silly and pointlessly arbitrary abstraction of marginal importance to them but it would be consistent with their knowledge of the way the universe works. So this is a scientific fact that I think we can say exists independent of humanity.

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.

⚛​

 

[CaptainQuasar]

Mathematical truths exist be for we "find" or prove them but not before we define them.

Do you really think so? That would seem to mean that if, a million years before humans even existed, some alien on the other side of the universe just happened to mathematically define the same thing we call a triangle in Euclidean 2-space (though perhaps from its perspective Euclidean 2-space would be an exotic and counter-intuitive concept), then because that's before we've defined triangles the alien might arrive at the conclusion that the sum of a triangle's angles is something other than 180°.

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.

⚛​

 

[Pythagorean]

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.

⚛​

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.

 

[Pythagorean]

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.

⚛​

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.

 

[CaptainQuasar]

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.

⚛​

 

[Pythagorean]

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.

⚛​

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.

 

[CaptainQuasar]

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.

⚛​

 

[Pythagorean]

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

 

[CaptainQuasar]

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.

⚛​

 

[Damir]

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,

 

[Georgepowell]

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

 

[CaptainQuasar]

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...

⚛​

 

[Chrisc]

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).

 

[Pythagorean]

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:

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.

 

[Georgepowell]

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"

 

[Pythagorean]

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.

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.

 

[Pythagorean]

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.

 

[CozmicScott]

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?