Mathematics

Math is such an awesome and beautiful phenomenon. It can compact volumes of physical meaning into a few symbols. It seems to be the best way to translate our physical experience of nature into descriptive meaning, and even seems to be an intrinsic part of it. In this thread I want to discuss the philosophy behind math, what it is, where it comes from. Is it inherent to external nature or do we impose it? If it’s a product of our neurology, is it then itself a part of “external nature” as a neurological structure as opposed to some disconnected abstraction? Are it's contents disconnected abstractions when describing other aspects of nature such as physics? When Pythagoras discovered mathematical laws about idealized right triangles, was he really uncovering a natural structure of neurology that creates idealized geometry? Do other life forms do math? For example, when a mother bird goes hunting and knows how many trips to take depending on how many chicks she has, or when a squirrel knows how many nuts to burry in the winter for it to survive, or does a blurry line between math and instinctual intuition happen here? These are just a few questions to start out with.

A question I can't answer is why reality outside of human scales seems to be based upon mathematics. I tend to view all human mental constructs as approximations of patterns which will break down when taken outside of the realm in which we formulated them (for instance, our conception of space and geometry as being rectilinear breaks down at scales outside of our experience, yet before knowledge of this many would have claimed that such a conception was 'inherently obvious' and 'an intrinsic part of reality'. Even more fundamental constructs like cause and effect have the same problem).

Yet with mathematics it was totally backwards, in one specific circumstance: we initially came up with complex numbers (two-dimensional numbers involving the square root of minus 1) more than two centuries ago when trying to solve totally prosaic problems (namely cubic equations, which can correspond to real questions about volumes, etc.), but all they were was an intermediary step on the way to the real answer which corresponded to something physical. These numbers turned out to provide a lot of insight into things like polynomials and limits (which were abstracted from intuitive, physical things), even though they themselves were originally an abstraction not corresponding to anything real.

The really weird thing is that quantum mechanics turns out to be pretty much intrinsically based on them. It's all formulated in complex numbers, and the underlying mechanism is based on how these numbers behave; whenever you want an answer, you end up measuring the 'size' of the underlying complex numbers (the size being a normal real number giving a real answer), yet the engine underneath the vehicle is all based on complex stuff.

Why on Earth is it that an obscure academic abstraction that came purely from consideration of macroscopic intuitions ended up being the basis of reality on the most fundamental scales, totally removed from the macroscopic world? How did we find the basis of reality without first having to explore down to this level??

What's always captured my wonder is this kind of ambiguity between math and physical reality. Sometimes it almost seems to me that you can say physical reality is math, or that it is an intrinsic part. Matter and energy come in discrete quantized forms which can be added, subtracted, multiplied, ect. If you add or subtract 1 electron from an atom, it will exhibit specific properties. A charged particle exhibits the behavior of being acted on by 1/4 of it's original force when it is twice as far away from an opposite charge. This seems to be much more than merely a description, these patterns and ratios seem to be the organization of nature.

The thing with that is, there has always been clouds of mystery on the horizon of our best conception of reality and our mathematical description of it. Space was well described by Euclidean geometry until Einstein unraveled one of the "two small clouds of mystery" Lord Kelvin mentioned, and completely changed the mathematical description of space to a geometry which Bernhard Riemann discovered purely for math itself. Now, even the reality of this (mostly) working description of space seems to be obscured by two clouds on the horizon, dark matter and dark energy. Does our best mathematical description of reality only work when we sweep those well contained and isolated areas of mystery under the rug? Will mathematical sureness always flow out into something new when those mysteries are uncapped?

Does a paradox between sure mathematical description and the unknown forever reform and evolve science? Will it eventually end in sure mathematical description (is reality at it's core mathematical organization?), or will we just use our best mathematical structures as stepping stones to forever ascend into new mysteries (is reality at it's core the ineffable unknown?). If the latter is true, where is the math coming from if not from physical reality? Is it something uniquely neurological? Are we just projecting neurological structures onto the rest of the observable universe? However, our brains are constructions of the universe: physics, chemistry, ect, so surely whatever our brains do must be akin to properties of the rest of nature, perhaps a kind of primordial math of which we discover being an intrinsic one, this would seem to make the former possible. Could it be possible that an intrinsic part of human experience is that both are true in a conserved dual awareness kind of way? This has been in the back of my mind forever lol.

Quantum Mechanics is a prime example of this kind of ambiguity. Is math really the core of this reality we observe, or is it mystery? There is heated debate on both sides.

I'm not really sure how the discreteness of electron energy levels suggests anything about mathematics. In one version we have that the angular momentum is anywhere on the real line; in another we have that the angular momentum is anywhere on the natural line (scaling for the Planck constant). Why do you think discrete stuff is more mathematical than continuous stuff?

Just the fact that Planck's constant is like the integer of nature makes it seem like algebra is nature instead of just a description of it. I think the fact that I just started calculus made me bias and unfamiliar with thinking about infinite concepts mathematically. Even with continuous stuff though, we only ever have an incomplete description that points toward an intangible reality, which makes me think it's really just description or that maybe they embody a kind of math yet to be discovered. Math seems to me to be inherently discrete, even when describing continuous things, because the actual continuity is only ever pointed to, leaving you to your imagination. (correct me if I'm wrong, I'm interested.)

I've always liked to think of mathematics as the "code" that tells the universe how to behave; a series of interdependent lines of "code" or even just a single line that every other physical law in the universe is derived from. It would have to be, considering the universe itself was theoretically created at a single instant in time (whether it be from the big bang or Divinity) which must have, itself, been adhering to some uniformity. This would be making deterministic assumptions, however.. that with the right formula and "given" measurements we could predict the arrangement of all matter in the universe at any moment in time. Free will would be lost.

I'm not sure I really understand the idea of mathematics being something separate from the universe that "tells it what to do", it seems to me it either has to be something that describes the universe with increasing accuracy, or an inseparable quality of the universe.

The big bang is our best guess so far I suppose, but given that it was itself a quantum event that gave birth to the quantum and macroscopic physics that we see, I find it difficult to see how it could be described fully by a mathematical code without some form of new mathematics or a different conception of quantum physics.

I see what you're saying. When I say that mathematics is the "code" of the universe, what I really meant is that mathematics is our most accurate means of defining or touching upon the "code". Most mathematical formulas (particularly in calculus) are theoretical and thus deal with abstract quantities and ideal situations. And because matter operates on a submicroscopic we can never really know whether or not the fundamental particles of the universe's motion is in harmony with even our most basic formulas. I guess the accuracy is more or less assumable being that mathematics is derivative of the science of logic. So unless humans have some sort of delusional way of observing and being in the universe, there should be no reason why we can't trust mathematics to be an accurate translation of the "code" of the universe into terms we can conceptualize.

I understood what you were saying at first then you went the opposite way in the last few sentences. First you said we can't really know if the universe follows are most basic formulas, then you say we must trust them or be delusional. Well these seem to be the absolute ends of the ambiguity I was talking about, I think we may be able to trust math for our best conception of reality, but who's to say that it won't evolve into different mathematics when the mysteries are uncovered, which have been consistently ever present no matter how much we reform our conception of nature more accurately. I'm not excluding the idea that we will someday reveal nature as a purely mathematical structure, though it's certainly not going to be with the conceptions we have now, and I wouldn't call it delusional to seek a new logical way to do this.

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Originally Posted by Xei

A question I can't answer is why reality outside of human scales seems to be based upon mathematics. I tend to view all human mental constructs as approximations of patterns which will break down when taken outside of the realm in which we formulated them (for instance, our conception of space and geometry as being rectilinear breaks down at scales outside of our experience, yet before knowledge of this many would have claimed that such a conception was 'inherently obvious' and 'an intrinsic part of reality'. Even more fundamental constructs like cause and effect have the same problem).

Yet with mathematics it was totally backwards, in one specific circumstance: we initially came up with complex numbers (two-dimensional numbers involving the square root of minus 1) more than two centuries ago when trying to solve totally prosaic problems (namely cubic equations, which can correspond to real questions about volumes, etc.), but all they were was an intermediary step on the way to the real answer which corresponded to something physical. These numbers turned out to provide a lot of insight into things like polynomials and limits (which were abstracted from intuitive, physical things), even though they themselves were originally an abstraction not corresponding to anything real.

The really weird thing is that quantum mechanics turns out to be pretty much intrinsically based on them. It's all formulated in complex numbers, and the underlying mechanism is based on how these numbers behave; whenever you want an answer, you end up measuring the 'size' of the underlying complex numbers (the size being a normal real number giving a real answer), yet the engine underneath the vehicle is all based on complex stuff.

Why on Earth is it that an obscure academic abstraction that came purely from consideration of macroscopic intuitions ended up being the basis of reality on the most fundamental scales, totally removed from the macroscopic world? How did we find the basis of reality without first having to explore down to this level??

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It's a red herring to say that complex numbers are intrinsic to quantum mechanics. You could do all of it with vectors. Complex numbers are just more convenient. Furthermore, if they were essential to quantum mechanics, then that still doesn't make them the "basis of reality on the most fundamental scales". It would just mean that they are an intrinsic part of one part of one model of reality that happens to be very interesting and occasionally useful.

So you're making a strong assumption in claiming that we've discovered the "basis of reality" without even having to explore it. Hilbert space happened to be a convenient model because all the solutions for various wave equations live in a hilbert space. With other equations it would be a symplectic manifold or something.

What it boils down to is that the space of mathematical systems is just so large that it would be very surprising to find a system that couldn't be modeled to any degree of accuracy using it. The vast majority of mathematics is only ever used indirectly (if at all) in modeling reality.

Again, no big thing. The other way around would be the surprise.

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Originally Posted by PhilosopherStoned

It's a red herring to say that complex numbers are intrinsic to quantum mechanics. You could do all of it with vectors. ...

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Thank you. I was going to point out the same thing but getting into an argument with Xei (who we all know has issues with admitting he's wrong) was not something I had the energy to do. Good on you, sir.

The OP mentioned Pythagoras. That's interesting, because Pythagoras is the perfect example of why mathematics is indeed a series of mental models, not reality. You cannot find a perfect right triangle in this universe. But, you can find situations where using certain geometrical rules is beneficial. For example, I would imagine that civil engineering would have been *slightly* more difficult without Pythag.

Another line of evidence is the fact that the mathematics get more complex, not simpler, as the physics gets deeper. This is consistent with humans trying to impose mental models on aspects of reality that are further and further removed from regular experience.

Uh yeah, 2D 'vectors' equipped with a multiplication rule isomorphic to that of complex numbers.

Otherwise known as... complex numbers.

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Originally Posted by PhilosopherStoned

What it boils down to is that the space of mathematical systems is just so large that it would be very surprising to find a system that couldn't be modeled to any degree of accuracy using it. The vast majority of mathematics is only ever used indirectly (if at all) in modeling reality.

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But we're not talking about obscure systems with layers upon layers of complicated rules and refinements. We're talking about a very simple structure that was already ubiquitous to mathematicians due to its potency in pure maths research - a structure discovered by extending abstractions from the familiar physical world (like the axioms of algebra) to realms with no physical meaning in that world, which turned out to be the natural language of phenomena entirely outside of human experience.

Stream of consciousness, I guess. Haven't been on a forum in quite some time so I'm still working out the kinks in my writing style. Interesting proposal though, that mathematics is as ephemeral as anything else in the universe. A wise man once said, "The only thing constant is change."

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Originally Posted by Xei

Uh yeah, 2D 'vectors' equipped with a multiplication rule isomorphic to that of complex numbers.

Otherwise known as... complex numbers.

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Vector and tensor equations are themselves just short hand notations for dealing with multiple equations. So otherwise known as ... "equations". Those have been around for a while. Again, it's just a choice of model.

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But we're not talking about obscure systems with layers upon layers of complicated rules and refinements. We're talking about a very simple structure that was already ubiquitous to mathematicians due to its potency in pure maths research - a structure discovered by extending abstractions from the familiar physical world (like the axioms of algebra) to realms with no physical meaning in that world, which turned out to be the natural language of phenomena entirely outside of human experience.

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You're using a lot of unclear words here. When you say the "natural language" of phenomena entirely outside of human experience, you do understand that you are assuming that

1) phenomena outside of your experience exist in any meaningful sense of the word
2) those phenomena come equipped with a "natural language" which is accessible to us.
3) we have actually discovered that "natural language".

So "natural language" may be a very confusing phrase here.

Furthermore, if you look at quantum physics you will see that it is an obscure system with layers upon layers of complicated rules and refinements. The fact that complex numbers are simple and ubiquitous in mathematics (they're pretty much determined once you have the integers if you restrict yourself to archemedean ring extenstions) is just an indication that they are expected to be useful in many mathematical models. QP happens to be one such model. What's the big deal?

Also, you're painting a very rosy picture of the invention of complex numbers. The axioms of algebra were ironed out in part to cope with the subsets of complex numbers (various "integral domains", e.g. the "Eisenstein Integers") which predate the development of algebra as an axiomatic field.

Finally, it's not like QP actually uses properties that are unique to the complex numbers like holomorphic functions and the like. Nothing depends on anything being holomorphic. In the context of QP, complex numbers are just convenient and glorified vectors. :)

math is a concept of representing things in numbers. or other word try to represent a complex system into a simpler notation. once summation is a bunch of + symbols, now got the idea to represent it with a single symbol of capital sigma, it goes on and on to higher level, integration, differentiation and whatnot. mathematicians also have device methods to make mathematical operation simpler, matrices, complex and imaginary numbers, alot of variety of spaces, domain and set up to fancy quaternion to process higher order of dimension. it is an intangible tool invented by human and keep evolving in space and time, just like the rest of invention (tangible or intangible). and the idea of the invention in the first place, implanted in the brain of the first man by "you know who".

ps: we think math is our marvelous method of solving probelms, but once a mathematician (cannot remember who or where) said something like... solving problem is only a small part in mathematics, the bigger part is finding a good symbols.

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Originally Posted by Volition

I see mathematics as more a way we've invented for conceptualising patterns and quantity relationships we observe in nature. For example, E=MC2 doesn't mean that the world is math or that math is an integral part of the world; it's just a quantity relationship we express and quantify for our own understanding using mathematics.

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But aren't quantity relationships in nature already quantified without us by definition? Are you saying that mass doesn't convert into energy in the exact ratio of E=mc^2 and that it's only our best description of their somehow essentially intangible quantities?

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Originally Posted by Wayfaerer

Are you saying that mass doesn't convert into energy in the exact ratio of E=mc^2 and that it's only our best description of their somehow essentially intangible quantities?

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E=mc^2 is not true to the last subatomic particle, and it's not true if anything is moving. It's an approximation like anything else.

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Originally Posted by Volition

I'm saying that mathematics is a way of us presenting the concept in a way we can understand. The ratio is already there in nature, mathematics is just our way of expressing the ratio we observe in a form we understand. The metalanguage of quantity relationships as it were.

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It's just I'm having a hard time completely separating "the ratio is already there in nature" and the ratio is Energy = Mass multiplied by the square of the speed of light. Nature seems to "write" this equation for us in it's behavior, unless of course we find a better way to describe it in the future.

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Originally Posted by cmind

E=mc^2 is not true to the last subatomic particle, and it's not true if anything is moving. It's an approximation like anything else.

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Proof? The conservation of energy is one of those things I have a hard time seeing as not completely true.

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Originally Posted by Wayfaerer

Proof?

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Uh, proof of what? You want me to prove the Heisenberg Uncertainty principle?

If you want, I'd be interested to see how you summarize/interperate it, but it won't prove the conservation of energy incomplete.

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Originally Posted by Wayfaerer

If you want, I'd be interested to see how you summarize/interperate it, but it won't prove the conservation of energy incomplete.

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Edit: Bad day today...

Are we talking about thermodynamics now or SR? Different answers to each question. It turns out that the 1st law of thermo and E=mc^2 are both literally false, but true enough in aggregate.

1st law: it's a statistical law, first and foremost. Individual particles follow random walks and are free to arrange themselves in high energy states, but it becomes astronomically unlikely when you get up to 10^23 of them.

E=mc^2: Some assumptions that can break down, such as the homogeneity and continuity of space time. Also it assumes the object in question is perfectly rigid (one side isn't moving with respect to the other) and perfectly stationary (or if in motion, the motion is perfectly understood). Hard to have such assumptions at the small scales.

Then there's Hisenberg. One variant of the Principle says that delta(E)*delta(t) >= hbar/2 (forgive me if the constant term is wrong, it's been a while). It means we can't have perfect knowledge of the energy of a system over an arbitrarily short time span. So that blows E=mc^2 and every other equation in physics out of the water, if interpreted literally. They're still true in emergence, but not in the details, understand? For example, we know that the hypothesized 'quantum foam' violates E=mc^2 in this regard. Yet over time it still tends to E=mc^2.

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Originally Posted by cmind

1st law: it's a statistical law, first and foremost. Individual particles follow random walks and are free to arrange themselves in high energy states, but it becomes astronomically unlikely when you get up to 10^23 of them.

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The first law of thermodynamics is that energy can change forms but cannot be created or destroyed, this is statistical?

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Originally Posted by cmind

Also it assumes the object in question is perfectly rigid (one side isn't moving with respect to the other) and perfectly stationary

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So what happens to non rigid objects? Their mass can convert to more or less energy than the equation says? Does it not compensate for motion?

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Originally Posted by cmind

Yet over time it still tends to E=mc^2.

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Right, I was under the impression that time-energy uncertainty of the vacuum still conserved energy over time.

I may be straying from the topic here but it seems like all the people that are trying to argue that mathematics is inherent in nature are forgetting one thing.

Reality is the bare perceptions that we experience. That's it. As soon as we put thoughts or words on it we are dealing with a model of reality and not reality itself. So if we choose to use mathematics to model some portion of reality, there should be no surprise when it appears that mathematics is fundamental to nature.

If we were to model reality using some other system and then confuse that model with reality we would think that that system was fundamental to nature. That's really all that's going on.

I wouldn't agree that reality is our bare perceptions, I believe our perceptions are a part of it, but that they are only a surface sheen of the rest of it. I get what your saying about words, the parts of reality aren't the labels we give them, but math seems different to me. The H20 molecule is composed of 1 hydrogen atom and 2 oxygen atoms. This isn't just our creation, this is what we observe nature does. If there is no one there to label this configuration as 1 hydrogen + 2 oxygen, does the reality of water become indescribable magic or something? The actual number 2 that you see on the screen right now isn't what I'm talking about, I'm talking about the actual reality of atoms, the concept of 2 seems to be real without us, I think we discovered it. Volition said something about a jaguar not being able to equate E=MC^2 as proof that the equation is not a part of nature itself, but a visual diagram or, even better, a visual observation of nature, would still be what the equation represents. I'm not defining math as the symbolic equations themselves, but the relationships they point to, which look a lot like nature. The jaguar might not have the capability of discovering the relationship between energy and mass, but if it's hungry it may fight one with food and back off if a family member comes so the hostile side becomes 2.

Just to playfully explore the idea I want to make an analogy regarding math and natural language. A natural language would be a verbal form of communication like English, French, etc. whereas math, logic and computer programming are formal languages. Both operate and derive meaning in slightly differently ways but are fundamentally similar.

A word (like r-o-c-k) is the symbolic relation of an object in the world (an actual rock) or a relationship between objects (like verbs or adjectives[hard, gray, kicked, etc]). So a word is just a mental construction that exists as a relationship between the subject (the observer) and the object (a rock). The word r-o-c-k is not an inherent part of the universe and would not meaningfully exist without human beings to understand it.

Similarly numbers are just symbolic representations of patterns, objects, relations, etc. in the world. As PhilosopherStoned said, math is a model of reality not reality itself. The patterns that math expresses would still exist no matter what arbitrary symbols we use to describe these patterns; just as a rock would exist no matter what words we use to describe it. The patterns do exist independent of the numbers and words used to describe them but not the other way around. Saying math is inherent in the universe implies that it exists independent of the human mind but it makes no sense to deny that math only exists as a relationship between the observer and their environment.