Misunderstanding basic math concepts, help please?

[rmsgrey]

Yes that is what I mean. In the end "absolute truth" is two words and I define them to mean this particular thing. I take this meaning because I think it is the meaning most people intuitively associate with those words when they hear them.

Most people intuitively expect to have it both ways:

On the one hand, people expect absolute truths to exist objectively and independently of human thought. On the other hand, people expect absolute truths to give them 100% certainty.

That's exactly the crux of the problem.

BTW I think that this is one of the reasons (not the only one, of-course) that revelation-based religions are so popular. If we assume, on a leap of faith, that the words in some Holy Book are absolute truths, we can have it both ways (or at least delude ourselves that we can).

It's also somewhere near the heart of Descartes' follow-up arguments to the cogito - he goes through considerable contortions to try to convince himself that his being convinced of something is sufficient evidence for it to be true by, essentially, invoking divine revelation - there is a god (by handwave) and since god is good, he would not permit humans to be completely deceived about the world around them, so the mere fact of being convinced the world works a certain way is evidence that god wants you to believe it, and so that it must be true...

Personally, I take a more modest position - I recognise that there are certain things I believe absolutely, and cannot argue myself out of believing, so I might as well accept them as true and explore their consequences while conceding intellectually that it might not be the actual truth. This way, I justify my belief in things like the internet and this post existing, without having to prove it real or invoke divine revelation - I believe it because I cannot believe otherwise, so I might as well move on.

[Cleverbeans]

In what way is math "vague"? Imperfect - I agree. Nothing is perfect. But vague?

Yes vague, as in only to a certain degree of accuracy. In the way that say, a quadratic can be made to vaguely resemble part of a sine curve or how we study non-linear curves by pretending they're close to linear on a small region.

Also, the laymen revere math? Math skills are socially rewarded? That is a very strange land of which you speak. Never seen such a place in my life, unfortunately.

Well I'm in Canada which is arguably a strange land. Mathematical ability was praised and encouraged starting in grade school, but maybe my teacher were the exception?

Perhaps you meant to say that the laymen believe that the mathematicians themselves think math is perfect? Because in a way, that's the basis for what Treatid is arguing here: "you guys think math is perfect. I'm the only person who accepts the fact that it isn't".

Treatid thinks that but I don't really know anyone else making such a claim. I've seen this sort of unwavering believe in conspiracy before but it's typically a mental health issue like delusional disorder or schizo-effective. I don't think it's common in the general population.

[PsiCubed]

Yes vague, as in only to a certain degree of accuracy. In the way that say, a quadratic can be made to vaguely resemble part of a sine curve or how we study non-linear curves by pretending they're close to linear on a small region.

I know what the word "vague" means, but I still don't understand what you're getting at.

How is mathematics "vague"? How are statements like 2x3=6 or "11 is a prime number" or "there does not exist a one-to-one correspondence between the set of integers and the set of real numbers" in any way vague?

Well I'm in Canada which is arguably a strange land. Mathematical ability was praised and encouraged starting in grade school, but maybe my teacher were the exception?

Oh... you mean that the teacher was praising and encouraging mathematical ability?

That makes sense. I thought you meant that this is the general attitude of the public.

Anyway, speaking of that teacher:

Did he ever bother to explain why he is saying that "mathematics is the universal language"? Did he give examples? And more importantly: did he give examples which aren't part of the school learning material?

I'm asking this, because high school math teachers seldom know enough about math to substantiate such grand claims. Most math teachers, just like the general public, think that "math" is just the stuff they teach at school: arithmetics, algebra and basic geometry. And let's face it: school math is tedious and boring and uninspiring. Trying to call that "the universal language" is a wonderfully pompous and baseless proposition.

It's also somewhere near the heart of Descartes' follow-up arguments to the cogito - he goes through considerable contortions to try to convince himself that his being convinced of something is sufficient evidence for it to be true by, essentially, invoking divine revelation - there is a god (by handwave) and since god is good, he would not permit humans to be completely deceived about the world around them, so the mere fact of being convinced the world works a certain way is evidence that god wants you to believe it, and so that it must be true...

Personally, I take a more modest position - I recognise that there are certain things I believe absolutely, and cannot argue myself out of believing, so I might as well accept them as true and explore their consequences while conceding intellectually that it might not be the actual truth. This way, I justify my belief in things like the internet and this post existing, without having to prove it real or invoke divine revelation - I believe it because I cannot believe otherwise, so I might as well move on.

I'll go even further:

Let's say we did manage to believe otherwise. What then?

The biggest problem with doubting that 2+2=4 or thinking that reality is an illusion, is that such doubts are completely useless. Okay, so everything is a lie. What are we to do with this kind of information?

It doesn't tell us anything useful. It doesn't guide us in any way or help us grow as human beings. It is a kind of philosophy that self-destructs by implying that philosophy itself is pointless.

So believing in that is synonymous to giving up. Why would anybody want to do that?

[gmalivuk]

In what way is math "vague"? Imperfect - I agree. Nothing is perfect. But vague?

Yes vague, as in only to a certain degree of accuracy. In the way that say, a quadratic can be made to vaguely resemble part of a sine curve or how we study non-linear curves by pretending they're close to linear on a small region.

There's no pretending, smooth curves *are* close to linear on a small region, in a very precisely defined way. ("Precisely" being, of course, the exact opposite of "vaguely".)

Statements like that mean very specific things when mathematicians say them, and I can't help but think you know very little about math if you think that makes it vague.

[Cleverbeans]

Statements like that mean very specific things when mathematicians say them, and I can't help but think you know very little about math if you think that makes it vague.

I realized after I went to bed I didn't explain myself very well. By vague I mean more that when mathematical models are applied to empirical data that it's a lossy. Line of best fit Gaussian optics would be a better example. We give up some precision to find a pattern and when applying that model to make predictions we expect a certain amount of error. I hope that makes more sense.

[gmalivuk]

Ah, so when you said "mathematics is vague", what you actually meant was "physicists use approximations"?

[Cleverbeans]

Ah, so when you said "mathematics is vague", what you actually meant was "physicists use approximations"?

Yes. Vague only as a description of the world but not internally. I agree that on it's own mathematics is very precise.

[PsiCubed]

Ah, so when you said "mathematics is vague", what you actually meant was "physicists use approximations"?

Yes. Vague only as a description of the world but not internally. I agree that on it's own mathematics is very precise.

But that isn't really true, either.

The basic laws of physics are governed by very simple mathematical rules. Or at least, we have no reason to assume otherwise.

The problem is, that for anything beyond the simplest of physical systems, using the basic laws of physics is not practical. So we make simplified models which approximate the behavior of the exact solution.

For example, take the trajectory of a baseball. Instead of tracking the movements and interactions of a trillion trillion atoms, we can treat the ball as a rigid body and the air around it as an ideal gas fluid. Now, with these assumptions (which are physically inaccurate) the problem becomes simple enough to be mathematically tractable.

For better accuracy, we can add more factors (say, a simple model of the effects of air pressure on the shape of the ball). But whenever we do that, we are changing the actual physical system we are talking about. Nobody seriously doubts that if we did write the trillion trillion equations needed to completely describe an actual flying baseball, we'd get a completely accurate result.

In short:

Any "vagueness" you preceive here is just a result of these practical limitations.

[Zohar]

Nobody seriously doubts that if we did write the trillion trillion equations needed to completely describe an actual flying baseball, we'd get a completely accurate result.

With the only caveat being we don't have a perfect understanding of the physical laws governing all those particles.

[Cleverbeans]

The basic laws of physics are governed by very simple mathematical rules. Or at least, we have no reason to assume otherwise.

It's also an assumption that this is true too, so we're assuming either way. I highly object to the term "governed". The math doesn't control the universe, it describes it based on our limited observations. That's sort the beauty of the empirical sciences - our ability to overturn our previous models when presented with better information.

The baseball example is apt. We have to ignore a great deal of detail in the model to find something useful to describe it's behavior. Even when we have a strong empirical model we often can't use that model to solve even relatively small systems like say, the three body problem can be incredibly complex, even intractable. Our inability to reconcile our theory of gravity with quantum mechanics is another case where the math breaks down as a consistent description of the universe. There seems to be an expectation that these forces can be resolved into a universal theory but I'm not certain we have any reason to believe that. I understand the historical context that motivated unification and why it was so successful but it seems plausible to me that this can reach a hard limit. I believe this is because of the same tacit assumption that led you to use the word "governed" earlier. The math has proven to be a powerful way to describe these specific things and we've come to have faith in it's power, but perhaps too much.

Edit:

Nobody seriously doubts that if we did write the trillion trillion equations needed to completely describe an actual flying baseball, we'd get a completely accurate result.

I certainly doubt this. The halting problem is a rather obvious example where no matter how many equations we have we know there is no mathematical method to solve this problem. Godel's theorem tells us there are true, unprovable statements even when we don't try to fit models to the real world. There are serious limits to it's power.

[PsiCubed]

The basic laws of physics are governed by very simple mathematical rules. Or at least, we have no reason to assume otherwise.

It's also an assumption that this is true too, so we're assuming either way.

Okay, let me rephrase my statement:

The basic laws of physics, as we know them, are governed by very simple mathematical rules.

The fact that we may be mistaken regarding these laws is irrelevant. We could be talking about Newtonian mechanics, which we already know to be inaccurate in the real world, an my point would still stand.

I highly object to the term "governed". The math doesn't control the universe, it describes it based on our limited observations.

It's just a figure of speech. Of-course math doesn't "control" the universe. Math is just a language, and a language can't control anything.

But you gotta admit that a language that can sum up everything we know about the way the universe works in a few pages, is pretty powerful.

That's sort the beauty of the empirical sciences - our ability to overturn our previous models when presented with better information.

Yet, at the fundamental level, this doesn't happen very often.

Physicists don't change their mathematical models of the world every Thursday. Had they done that, I would have conceded your point.

But they don't. General Relativity has been with us for exactly 100 years now. Quantum Field Theory - in its present form - is about 50 years old. This is pretty impressive, when you consider how many new experiments have been done in the past few decades.

Sure, sometimes the universe surprises us. This has absolutely nothing to do with the limitations of mathematics as a language, and everything to do with the fact that science works by the imperfect process of logical induction.

We don't know everything. Whenever we create a scientific model, we're just making an educated guess. So naturally, we make mistakes regardless of the "language" we write our guesses in. Given these limitations, it is nothing short of miraculous that our mathematical guesses about the cosmos needs to be updated only a few times every century.

Even when we have a strong empirical model we often can't use that model to solve even relatively small systems like say, the three body problem can be incredibly complex, even intractable.

The intractability of the three body problem has absolutely nothing to do with math. It has to do with the simple fact that the position of the bodies after a very long time is incredibly sensitive to their starting positions.

So unless we know that initial state with infinite accuracy, there's no way - in principle - to differeniate between two very similar initial states that would diverge in the long run.

This is true regardless of the method we try to use to predict the future of the system. So once again, this isn't a weakness of mathematics but an inherent complexity of the system itself.

Our inability to reconcile our theory of gravity with quantum mechanics is another case where the math breaks down as a consistent description of the universe.

We already have a mathematical model which reconciles the two: String Theory.

And this model actually serves as another example of the power of mathematical language: The original idea of String Theory wasn't meant to include gravity at all. It was originally proposed to solve a minor inconsistency in quantum field theory... but once the math was worked out, it turned out that String Theory predicts the existence of gravity! To be precise: It predicts a massless gauge boson of spin 2, which is exactly the QM description of the graviton.

Of-course, without hard physical evidence, there's no gurantee that this beautiful theory has any connection with the real world. And right now we don't have the technology to obtain such evidence - either way.

At any rate, regardless of whether the model is physically valid, there's no question that the language of mathematics - via String Theory - is strong enough to bridge the gap between QM and gravity.

There seems to be an expectation that these forces can be resolved into a universal theory but I'm not certain we have any reason to believe that. I understand the historical context that motivated unification and why it was so successful but it seems plausible to me that this can reach a hard limit. I believe this is because of the same tacit assumption that led you to use the word "governed" earlier. The math has proven to be a powerful way to describe these specific things and we've come to have faith in it's power, but perhaps too much.

Perhaps.

We should certainly be careful whenever we try to make blanket statements about how things will turn out in the future. But right now, given the limited knowledge we have today, math really does seem to be "the universal language".

Nobody seriously doubts that if we did write the trillion trillion equations needed to completely describe an actual flying baseball, we'd get a completely accurate result.

I certainly doubt this. The halting problem is a rather obvious example where no matter how many equations we have we know there is no mathematical method to solve this problem.

Huh? What does the halting problem has to do with the issue?

It is obvious that the thing I've said is impossible in practice. But in principle, there's nothing to stop you from solving ANY set of differential equations to ANY desired accuracy.

The halting problem, on the other hand, is undecidable precisely because it is an infinite problem (if a program never halts, you can't determine this in a finite time. At least not in the general case).

Godel's theorem tells us there are true, unprovable statements even when we don't try to fit models to the real world. There are serious limits to it's power.

Nobody here claims that math has no limits. One doesn't even need to go the obscure depths of Godel's work in order to see this.

For one thing, math can't describe the human condition or anything else which we don't have a formal accurate description of. Funny that you said earlier that math is "vague", where this is exactly where the chief weakness of mathematical language lies: Math doesn't know how to do "vague". All the wishy-washy thoughts that we humans have most of the time, are completely beyond any mathematical description. This is math's biggest blessing, but it is also it's biggest curse.

[Cleverbeans]

But you gotta admit that a language that can sum up everything we know about the way the universe works in a few pages, is pretty powerful.

On the contrary. We only have a few pages worth of phenomenon that can be described mathematically, that's strong evidence of it's limitations.

Yet, at the fundamental level, this doesn't happen very often.

True, however it often coincides with the development of additional mathematical tools. As the language expand and becomes more descriptive we can model more phenomenon, and do it more accurately than before. Again, I find this indicative of our imperfect perception of reality. It works because we developed the language to intentionally describe it the way we perceive it. That doesn't seem particularly profound to me.

[]Sure, sometimes the universe surprises us. This has absolutely nothing to do with the limitations of mathematics as a language, and everything to do with the fact that science works by the imperfect process of logical induction.[/quote]

How did you reach that conclusion? Applied mathematics is in many ways about developing the language to interpret these surprises. We have to expand mathematics every time this happens and sometimes it doesn't work at all.

We don't know everything. Whenever we create a scientific model, we're just making an educated guess. So naturally, we make mistakes regardless of the "language" we write our guesses in. Given these limitations, it is nothing short of miraculous that our mathematical guesses about the cosmos needs to be updated only a few times every century.

We invented the language specifically to describe the cosmos I don't see that as any more miraculous than painting a picture or building a scale model. If we build something to do a specific task, should we be surprised when it does that task?

The intractability of the three body problem has absolutely nothing to do with math. It has to do with the simple fact that the position of the bodies after a very long time is incredibly sensitive to their starting positions.

Another way to say this is that because of the instability mathematics is useless for making predictions here. It simply can't describe what's happening because it requires specific information we have no access to. You're essentially saying the universe is too complex to be described mathematically. I feel like you're making my case for me here.

We already have a mathematical model which reconciles the two: String Theory.

Not one supported by evidence. It's very easy to describe the universe if we just assume it's a certain way but it's not meaningful.

At any rate, regardless of whether the model is physically valid, there's no question that the language of mathematics - via String Theory - is strong enough to bridge the gap between QM and gravity.

That's your assumption but we have no reason to believe it. Just because we can imagine the universe is a certain way doesn't mean it is that way.

math really does seem to be "the universal language".

It occurred to me how ironic this claim is since even on Earth only a small minority actually know this language, but I digress. It's as you mentioned before, it's incapable of describing the three body problem. That doesn't seem very universal to me.

Huh? What does the halting problem has to do with the issue?

It demonstrates conclusively that mathematical methods are incapable of solving some internal problems that don't even rely on empirical data. It's not that important to my argument though.

Nobody here claims that math has no limits.

Then it's a question of how good those limits are. So far it only describes a few pages of information. Information theory here to show our knowledge must be a form of lossy compression. In some sense have to blur reality to contain it in out tiny minds.

we humans have most of the time, are completely beyond any mathematical description.

At this point I think we're working with different definitions of universal. In one sense it's universal since anyone could arrive at these conclusions by noticing the logical nature of causation and building from there. In another sense it's not universal in that most of the universe can't be described this way. Can we both agree on this or am I missing something?

[gmalivuk]

The universe may be too complex for mathematics to describe it, but the three-body problem doesn't demonstrate that.

A purely Newtonian universe, where Newton's laws *perfectly* describe the motions of massive objects, would still have a three-body problem. A very simple set of equations to describe the motion of literally everything in the cosmos, and you're saying that would prove how little use math actually is?

(Incidentally that also disproves your "lossy compression" comment. The equations that perfectly describe everything might fit in a very small space, even if the phenomena that emerge are incredibly complex.)

[PsiCubed]

On the contrary. We only have a few pages worth of phenomenon that can be described mathematically, that's strong evidence of it's limitations.

Huh? What the **** are you talking about?

Einstein's General Relativity describes the behavior of ANY gravitational system. The Standard Model describes the three other fundamental forces - completely.

There are countless physical systems which are simple enough to be directly calculated by the above. And countless more which can be described in decent accuracy (like that baseball example we've discussed earlier).

You call that "a few pages worth of phenomenon"? Really? If what you said were true, that we need a few pages of math to describe a few pages worth of phanomena, then math would have been completely and utterly useless.

True, however it often coincides with the development of additional mathematical tools.

Not as often as you think.

The math for Einstein's General Relativity was invented by Riemann 50 years earlier.

Group Theory, which revolutionized quantum mechanics in the 1960's, was invented over 100 years earlier (at least as early as Galois in the 1830's).

Of-course, sometimes it does happen the other way around. Newton inventing calculus, for example. But considering that calculus has so many diverse uses (most of which Newton never dreamed of), this is hardly supportive of your argument.

It works because we developed the language to intentionally describe it the way we perceive it. That doesn't seem particularly profound to me.

Think about it for a moment:

We look at the world around us, make up descriptions of what we see as we go along, and this gives us models which remain relevant for decades of new obseravations.

That doesn't seem profound to you? Boy, you're a very hard person to satisfy...

(and as I've already stated, even this miraculous description cuts mathematics short, because in most cases we've developed the mathematical language decades before it was relevant in any way to physics)

We invented the language specifically to describe the cosmos I don't see that as any more miraculous than painting a picture or building a scale model. If we build something to do a specific task, should we be surprised when it does that task?

When it works so well for 400 years, even after our view of the universe was revolutionized half a dozen times in the process: Yes, we should be surprised.

Especially when no other human language even comes close to having this kind of power.

You want to compare our model of the universe to a scale model? Fine. It is an animated scale model of the entire universe. A model which is run by a computer program shorter than a modest chess program, yet it is so detailed that it describes... everything. And while it is not perfect, it is good enough to give us correct results in 99.99999% of the cases.

Oh, and in that once-in-a-blue-moon occurence where the model needs updating? We somehow have all the needed new parts waiting at bay to be installed. These new parts weren't even built for this purpose. They were built by the kids next door who just love to play with stuff and build beautiful complex structures.

Most of these kids actually regard our litte scale model with scorn. "The stuff we build doesn't have to be useful!" is their cry. Yet somehow, whenever we need a new part of our scale model, one of their toys fits the bill perfectly.

But other than that, yeah. It's just a scale model. Nothing too impressive

The intractability of the three body problem has absolutely nothing to do with math. It has to do with the simple fact that the position of the bodies after a very long time is incredibly sensitive to their starting positions.

Another way to say this is that because of the instability mathematics is useless for making predictions here. It simply can't describe what's happening because it requires specific information we have no access to.

Oh come on...

Mathematics can easily describe what's happening in the three body problem. It even tells you - exactly - how accurate your initial measurements need to be in order to be able to predict the future up to time t with the desired precision.

We can't actually do that? Well, that's too bad. But it isn't math's fault. Don't kill the messenger just because you don't like the message.

You're essentially saying the universe is too complex to be described mathematically.

I'm essentially saying that the universe is too complex for us to know everything, let alone predict the future.

So yes, if you expect mathematics to be some kind of lamp genie that can grant you any wish, you'll be sorely disappointed. Math is a language and a tool, not some kind of voodoo magic.

We already have a mathematical model which reconciles the two: String Theory.

Not one supported by evidence. It's very easy to describe the universe if we just assume it's a certain way but it's not meaningful.

Easy? In your last post you've claimed that this very task of mathematically unifying QM and GR is "impossible". You've made a huge deal of the claim that "math breaks down" when we try to do that.

So I'm telling you that it doesn't. Math, as a language, can handle it.

As for the evidence: We don't have the technology to look for it. We have zero experimental data, so we are forced to make a guess. What does this have to do with the limits of mathematics as a language?

Nobody here claims that math has no limits.

Then it's a question of how good those limits are.

That's exactly right. This question is at the crux of this entire discussion.

So:

...So far it only describes a few pages of information...

Already refuted that at length. The real situation is exactly the opposite.

Information theory here to show our knowledge must be a form of lossy compression. In some sense have to blur reality to contain it in out tiny minds.

As gmalivuk already hinted, actual compression algorithms serve as a nice example of the things we've talked about here.

A typical photograph can be compressed by at least a factor of 2 WITHOUT ANY LOSS of information. English text can usually be compressed by at least a factor of 5, because of the patterns of the language.

The more pattern and order there is in a file, the better it compresses.

So, what does this have to do with our universe? Well, it seems to be full of patterns and repetition. There are about 10^90 particles in the universe, but only a few dozen different types of them, and only 4 basic types of possible interactions between them. The physical laws and the building blocks are the same everywhere. And this is exactly why our mathematical models work so well.

At this point I think we're working with different definitions of universal. In one sense it's universal since anyone could arrive at these conclusions by noticing the logical nature of causation and building from there. In another sense it's not universal in that most of the universe can't be described this way. Can we both agree on this or am I missing something?

I agree with your distinction of the two types of "universal".

But I can't agree with your claim that "most" of the universe can't be described mathematically. I grant you that SOME things seem to defy mathematical description, but these things are the exceptions and not the other way around.

And you know the really weird thing? It didn't have to be that way. I can't think of any a-priori reason to believe math would describe the world so well. Yet it does. So at least in our own universe, saying that "math is the universal language" seems like a reasonable thing to say.

[gmalivuk]

To boil a lot of that down (a bit lossily): If math can describe a few pages, then it can describe the whole universe, because the laws that apply to those pieces of paper apply to the entire universe. We may have no reason to expect such symmetry, but given the symmetries, the compressibility should come as no suprise.

[Twistar]

The point I want to make here is that I agree. Math is great, fantastic, wonderful beautiful, has helped raise the standard of living for billions of people around the world etc. However, I just think that claiming anything "absolute" or "universal" about it is just claiming too much. Heck, at the moment we don't understand dark matter or energy of which a large part of universe is apparantly composed. That's not to say I doubt math will eventually come to explain that, but who knows what mysteries are over the next horizon? It's a purely intellectual point, but I think it's important in this type of philosophical discussion. Just don't call it universal, or accessing absolute truth and I'll be happy.

Again, I emphasize the fact that you can say all of the substantive things you want to say about math without saying it is universal. You can say math does an amazing job describing our world. We can explain so many phenomena. And if there's a phenomenon that we can't explain, historically, we have always been able to eventually figure out a way! Wow that's amazing! And furthermore, with enough study anyone in the world can understand some particular mathematical statement because the language is so clear and unambiguous. This is all so amazing and makes me love math so much!

But when you say "math is a universal language" it's just an overpowered under descriptive praise of mathematics that only serves to confuse one who may not be as familiar with math as you are rather than enlighten them. Instead of saying "math is a universal language" just say what you mean to say. That or be painfully explicit about what you mean by "universal language" (which will result in you saying what you mean to say.)


And just a reminder, no one here doesn't think math is great. We got on this tangent because apparently there is a disagreement amongst the people in this thread over whether math can access "absolute truth". We got onto that issue because it is at the heart of what is bothering Treatid about axiomatic mathematics as he sees it. I argue against the universality of mathematics here because I think it might be very easy for him to see a statement like "math is a universal language" and jump to the conclusion "math communicates 'absolute truths'", where 'absolute truth' is used in the sense that we have all agreed in the past few posts that mathematics CANNOT access. So to this end myself and I think Cleverbeans are trying to pry on the few weaknesses of mathematics to show that they really are there and that there is in fact a gap between mathematics and perfection. We are trying to do this because we are trying to convince Treatid that people DO already acknowledge that mathematics already has the very weaknesses he is pointing out. My feeling is that statements like "mathematics is a universal language" undercut this effort.

edit: a few typos.

[gmalivuk]

Why do you and Cleverbeans both keep talking about the current state of physics to make claims about the limits of mathematics? Our not knowing everything about dark matter, dark energy, or quantum gravity has nothing to do with limits inherent to mathematics, and everything to do with our lack of relevant empirical data.

Though you're doing it in a different way than Treatid, you also seem to be attributing fantastic promises to math and then arguing that it fails to deliver, even though math doesn't make those promises.

[Twistar]

Why do you and Cleverbeans both keep talking about the current state of physics to make claims about the limits of mathematics? Our not knowing everything about dark matter, dark energy, or quantum gravity has nothing to do with limits inherent to mathematics, and everything to do with our lack of relevant empirical data.

Though you're doing it in a different way than Treatid, you also seem to be attributing fantastic promises to math and then arguing that it fails to deliver, even though math doesn't make those promises.

I'm not attributing fantastic promises to math. It seems like PsiCubed is attributing fantastic promises to math by claiming it can access absolute truth and that it is a universal language.

The only reason I brought up dark energy was just to give a blatant example of something in the universe that math doesn't describe. This is just to give empirical evidence (though empirical evidence is not needed) to my claim that we are not justified in calling mathematics a universal language.

Again, my point is purely semantic. Just don't say things like "math accesses universal truth" or "math is a universal language" and I'll be fine.

edit: removed my confusion about why physics ended up coming up at all.

[Cleverbeans]

Why do you and Cleverbeans both keep talking about the current state of physics to make claims about the limits of mathematics?

I believe this is because of the two distinct meanings of universal being used. I was discussing whether mathematical models are capable of describing everything in the universe and the other seem to be whether mathematical truths are accessible by everyone without any a priori understanding of the subject. So universally descriptive versus universally accessible. Since we're working with different definitions we ended up going in circles for a bit.

[gmalivuk]

Why do you and Cleverbeans both keep talking about the current state of physics to make claims about the limits of mathematics?

I believe this is because of the two distinct meanings of universal being used. I was discussing whether mathematical models are capable of describing everything in the universe and the other seem to be whether mathematical truths are accessible by everyone without any a priori understanding of the subject. So universally descriptive versus universally accessible. Since we're working with different definitions we ended up going in circles for a bit.

Who says "universal truth" to mean the first thing?

[Cleverbeans]

Who says "universal truth" to mean the first thing?

Truth? Lets not get carried away here I said universal language. I haven't done a poll or anything but I'm pretty sure I'm not the only using that definition.

[Twistar]

Clearly everyone is using their own definition of 'universal' and drawing their own distinctions in their own head with the end result being us all just talking past each other. I think we should just drop the terms altogether, since, as I've said before:
1) they don't add anything to the discussion (you can always use other words and be more descriptive)
2) they open up tons of room for confusion.

If you all disagree and want to keep using the terms then we need to define:
1) Absolute truth
2) Universal truth
3) Universal language

I've given definitions for absolute truth = universal truth = something about which you have complete epistemic certainty. We've all agreed that under this definition mathematics does not access absolute truth since there is no way for humans to access absolute truth (under this definition). I don't think anyone has defined what a universal language is.

edit: I guess cleverbeans tried to describe universal language as a language that anyone anywhere can understand regardless of background or something. It wasn't super clear and the context was confusing.

[rmsgrey]

I've given definitions for absolute truth = universal truth = something about which you have complete epistemic certainty. We've all agreed that under this definition mathematics does not access absolute truth since there is no way for humans to access absolute truth (under this definition).

As I've said before, that's more of a definition of "absolute knowledge" than "absolute truth" since truth is generally accepted as being independent of people's beliefs, while epistemic certainty requires belief. For example, if the standard assumptions about the universe are correct, then Earth didn't become a ball sometime in the last 3000 years when educated people started believing it. Rather, Earth has been ball-shaped for billions of years before humans existed to have an opinion on the matter.

Mathematics may or may not access (absolute) truth. What it doesn't access is absolute knowledge or absolute certainty.

To paraphrase Charles Dodgson: if you wish, you could define "black" to mean "diffusely reflecting all incident electromagnetic radiation" and "white" to mean "absorbing all incident electromagnetic radiation" and, while I might think it unwise, and not adopt those meanings in my own writing, I am bound to accept that those are the meanings that attach to those terms as you use them in the context where you have so defined them.

[gmalivuk]

Who says "universal truth" to mean the first thing?

Truth? Lets not get carried away here I said universal language. I haven't done a poll or anything but I'm pretty sure I'm not the only using that definition.

Perhaps but
1) Saying it can be used to describe the behavior of everything in the universe is not the same as saying we know how to use it thus.
2) Saying it can describe all of physics is not the same as saying it can just as easily describe everything else at every other level of abstraction.
3) Surely there's some sense in using "universal language" to mean the sort of thing that would render a "universal translator" obsolete, and in using it analogously to how we already use "global language".

Edit regarding 2: The rules governing Conway's Life and the rules governing chess are very different, and it would be silly to try to analyze the latter using only the former. But if you built a massive computer in Life that was programmed to play chess, it would also be silly to claim that the rules governing Life weren't still being applied at a fundamental level.

[Twistar]

I've given definitions for absolute truth = universal truth = something about which you have complete epistemic certainty. We've all agreed that under this definition mathematics does not access absolute truth since there is no way for humans to access absolute truth (under this definition).

As I've said before, that's more of a definition of "absolute knowledge" than "absolute truth" since truth is generally accepted as being independent of people's beliefs, while epistemic certainty requires belief. For example, if the standard assumptions about the universe are correct, then Earth didn't become a ball sometime in the last 3000 years when educated people started believing it. Rather, Earth has been ball-shaped for billions of years before humans existed to have an opinion on the matter.

Mathematics may or may not access (absolute) truth. What it doesn't access is absolute knowledge or absolute certainty.

To paraphrase Charles Dodgson: if you wish, you could define "black" to mean "diffusely reflecting all incident electromagnetic radiation" and "white" to mean "absorbing all incident electromagnetic radiation" and, while I might think it unwise, and not adopt those meanings in my own writing, I am bound to accept that those are the meanings that attach to those terms as you use them in the context where you have so defined them.

Ok, fair enough. So there's a distinction between absolute knowledge and absolute truth. So I guess I should clarify that we all agree humans/math don't access absolute knowledge. The question of absolute truth seems sort of irrelevant if there's no way for humans to access it, so who cares if it is 'out there' or not.

It's still not clear to me what anyone here means by "universal language" when they say it.

[gmalivuk]

There are mathematical truths which are inaccessible to human knowledge, as has already been explained. Is that somehow more okay if we don't call such truths "absolute"?

[Twistar]

There are mathematical truths which are inaccessible to human knowledge, as has already been explained. Is that somehow more okay if we don't call such truths "absolute"?

I guess I'm being a bit sluggish on the uptake here but could you re-explain or point out where someone explained that there are mathematical truths which are inaccessible to human knowledge? Maybe someone explained it early but was using "absolute" and/or "universal" and I got hung up on that. In my mind all mathematical truths are accessible to human knowledge. That is, humans can fully understand mathematics. There is no impediment.

And yes, regardless of your response to the above paragraph, it is better, at least to me, if we don't call these truths absolute.

[gmalivuk]

No, I never said anything about absolute or universal in the post about such numbers, you just missed it.

Questions like, "Is pi normal?" surely have a factual answer, but it's conceivable that we will never know (i.e. prove) the answer to that question.

[Twistar]

No, I never said anything about absolute or universal in the post about such numbers, you just missed it.

Questions like, "Is pi normal?" surely have a factual answer, but it's conceivable that we will never know (i.e. prove) the answer to that question.

Ah I see. Ok. Well here's how I view this sort of thing.

First there's a distinction to make about the set of "questions that we will never know the answer to". I see at least two categories of these types of questions.
1) Questions which are independent of the axioms. If you take as your axioms the ZFC axioms then it is possible to prove that 1) it is not possible to prove the continuum hypothesis as well as 2) it is not possible to disprove the continuum hypothesis. In other words the truth or falsehood of the continuum hypothesis is independant of the axioms we choose. I repeat the example I gave above

1) A [Prem]
2) A-> B [Prem]

question: Is C true or false? It is impossible to prove whether C is true or false, in fact, we can take C to be EITHER true OR false and not get any contradictions. C is independent of the premises I have proposed.
The important thing to say about questions in this 1st category is that it is misguided to supposed that these questions "surely have a factual answer". They in fact do NOT have a factual answer and this can be proven. Of course you can rephrase my last couple sentences slightly differently if you taken different meanings for words or you take different philosophical viewpoints, but I think the way that I am viewing it is the least ambiguous. It would definitely not be good to bring the words "universal truth" anywhere near these sorts of questions.

2) The 2nd category of questions are questions which don't fall in the first category but we still may never know the answer to because humans just will never be creative enough to figure out the right way to do the proof. In other words, a proof exists, we just can't figure it out. These are questions which "surely have a factual answer" we just can't figure it out.
I definitely wouldn't apply the label "universal truth" to the answers to these questions, because again, they are only true relative to the axioms of the system. relative is the opposite of universal or absolute. However, I would DEFINITELY call these mathematical truths. In fact I think "mathematical truth" is a GREAT name for these sorts of things, it's clear and descriptive. It captures the idea that it is only a truth relative to the set of axioms underlying the system in which we are working, i.e. math.

[gmalivuk]

I'm talking about specific well-defined real numbers, so your repeated references to CH don't apply.

I'm not sure how you can boldly proclaim that proofs must exist which we're just not creative enough to find, when one proof we have managed to find is that almost all real numbers are uncomputable.

[Twistar]

I'm talking about specific well-defined real numbers, so your repeated references to CH don't apply.

I'm not sure how you can boldly proclaim that proofs must exist which we're just not creative enough to find, when one proof we have managed to find is that almost all real numbers are uncomputable.

Well if the proof doesn't exist then the question falls in the 1st category, no?

[Cleverbeans]

3) Surely there's some sense in using "universal language" to mean the sort of thing that would render a "universal translator" obsolete, and in using it analogously to how we already use "global language".

I agree and I'm willing to concede that in this sense math is a universal language.

Well if the proof doesn't exist then the question falls in the 1st category, no?

No. Godel proved conclusively that there are true statements consistent with the axioms which can never be proven.

[Twistar]

Well if the proof doesn't exist then the question falls in the 1st category, no?

No. Godel proved conclusively that there are true statements consistent with the axioms which can never be proven.

Yeah ok, I suspected it might have been something like that. I guess I need to brush up on the incompleteness theorem again. Anyways, I would call these statements mathematical truths not universal truths.

edit:
And does universal language just mean a langauge any human can understand if they put enough effort into it? Then wouldn't any language be a 'universal language'?

Sorry for being dense and pedantic here.

[PsiCubed]

I'm not attributing fantastic promises to math. It seems like PsiCubed is attributing fantastic promises to math by claiming it can access absolute truth and that it is a universal language.

When I was talking about "absolute truth", I was talking about absolute platonic truths like 2+2=4 and not anything in physics.

Physics only came into the picture once Cleverbeans made the claim that math "can't describe most the universe", and this statemet is simply false.

The only reason I brought up dark energy was just to give a blatant example of something in the universe that math doesn't describe.

Funny that you mention dark energy in this context, because there's no problem describing the concept mathematically. Einstein already did that 100 years ago. He called it "the cosmological constant". The weird thing is that he introduced it for all the wrong reasons. Yet over 80 years later, the exact same idea (and the exact same mathematical construct) happens to be exactly what we need to describe dark energy.

The original cosmological constant basically describes a universe filled with a fluid for which the equation of state is w = -1. An ordinary ideal gas has w=0, and relativistic fluids have w=1/3. So as you can see, we can get a "cosmological constant" by simply changing the value of a familiar positive parameter to a value less than 0.

The catch is, nobody knows what the heck "a fluid with w=-1" means physically. We haven't seen anything like that before, so there's nothing in our previous experience to compare it to. But in the language of mathematics, it isn't too difficult to describe.

(The modern theory of dark energy actually allows for any value of w less than 0. The actual measurements, however, favor a value pretty close to -1. Once again, the universe seems to "choose" simpler mathematical behavior over a more complicated one.)

Anyway, sorry for getting side-tracked into physics again.

And does universal language just mean a langauge any human can understand if they put enough effort into it? Then wouldn't any language be a 'universal language'?

"Universal language" means that if different cultures develop it independently, they will all be speaking the exact same language and be able to understand one another.

And the language of mathematics is such a language. "2+2=4" means the same thing on earth and on some distant planet.

No. Godel proved conclusively that there are true statements consistent with the axioms which can never be proven.

You mean "which can never be proven from within any specific axiomatic system"

There's nothing to stop us from using stronger and stronger axiomatic systems to prove all true statements about numbers. Godel's theorem just tells us that there's no "universal" axiomatic system that will catch all these proofs.

BTW where did Treatid disappear?

[Treatid]

Here, and paying very close attention to the conversation. Lots of really good points being made - many of which I've got really strong opinions on. But right now it seems to me that I'm more likely to derail the conversation than contribute - and between you all, you are doing a brilliant job of covering some essential ground.

I absolutely will be sticking my oar back in shortly.

Edit: Twistar's question: Agree.

[Cleverbeans]

Physics only came into the picture once Cleverbeans made the claim that math "can't describe most the universe", and this statemet is simply false.

Oh is that so? Lets start a short list of things you can't describe mathematically to get your started and just to be pedantic I'm going to make them all terrestrial.

Love.
Aesthetics.
Morality.
The placebo effect.
Pop culture.
Musical tastes.
Yo mama jokes.
Vellichor.
Ambition.
Schizophrenia.
Acting.
Justin Beiber.
Pedagogy.
Creativity.
Child abuse.
Petrichor.
Twerking.
Silliness.
Boredom.
Religion.

Need more?

[gmalivuk]

You're making an unjustified leap from the fact that we haven't explained those things mathematically to the conclusion that we can't. We haven't explained those things from the most basic physical principles, but unless you're some kind of dualist surely you accept that they are all products of basic physics, no?

You can't (practically speaking) explain how my phone works using the rules of Life, but in theory a computer could be made in the Life universe that does all the same things as the computer in my phone.

[Cleverbeans]

You're making an unjustified leap from the fact that we haven't explained those things mathematically to the conclusion that we can't.

I picked examples I'm confident can't explained mathematically.

We haven't explained those things from the most basic physical principles, but unless you're some kind of dualist surely you accept that they are all products of basic physics, no?

I see no reason to make that assumption. Emergent properties exist and by definition can't be described from more basic principles. For example the taste of salt can't be derived from the taste of sodium and chlorine independently. It's something new that requires independent description. This implies to me that reductionist arguments are insufficient.

[gmalivuk]

That the taste of salt isn't the taste of sodium plus the taste of chloride could easily be inferred from a reductionist argument, it would just have to be an argument made by someone with a basic knowledge of chemistry rather than, apparently, by someone like you.

The composite number 21 plus the composite number 10 sum to the prime number 31, while the prime 5 plus the prime 7 is the composite 12. That doesn't mean the properties of numbers are some kind of mystical thing that can't be explained by "reductionist arguments", it just means compositeness and primality aren't the sorts of things preserved by addition.

[Cleverbeans]

That the taste of salt isn't the taste of sodium plus the taste of chloride could easily be inferred from a reductionist argument

The burden of proof is yours. I'll leave you to it.