Are there any maths that are not comprehensible to the human mind?

you mean math is comprehensible?

Algebra. Most definetely

This is a circular argument. If your aunt had balls, she would have been your uncle.

life as we know it.
this does not include forms of life we can not even begin to imagine. who knows what could evolve in a universe that was not suited for our type of life.

http://www.youtube.com/watch?v=tCKqj-2JXZg

and anyway, the universe was not fine tuned for life. life finetuned itself for surviving in a universe that is to 99.9999999999999999999% lethal.

and if such a universe is the best a so called god can do….

is there math uncomprehensible for a human mind?
if you speak of math as we know it, no, because all of the known math has been invented by humans.
the math of aliens however is another story.

I am soooo going to answer this in the morning when I’m a bit more…with it!—FGQ!!!!

Well I sure as hell don’t understand any of my roommates Discreet Math class, and (if you consider physics to be higher math) most of the theory behind the 4th-12th dimension…

@D353R7F0XX can tell you more

Most of them, as far as I’m concerned. :-p

The anthropic principle is actually the idea that even though its extremely statistically improbable it has to have happened at least once because one in a million things still have to happen that one time, and we (earth life) are the result and the proof.

That’s what I meant by a circular argument. It must be so, because it is so. Were it not so, things would be in an alternate state, and your aunt would have balls.

It’s rather hard (maybe impossible) to explain a universe in terms that don’t follow mathematics. If mathematics really is reducible to logic then the proposition that universes must follow mathematics is the same as the proposition that logic constrains the possible universes. The problem is we cannot explain things that don’t follow logic. Even if a universe existed that didn’t follow logic would we be able to perceive it or understand it? It’s possible that some other form of logic exists that we don’t know. Since we are a product of the mathematics that compose our universe I can’t say that it’d be possible to comprehend this other form of logic (if it existed).

It seems entirely possible that there might be such incomprehensible universes. It might be useful to ask the question “Can illogical universes exist?”

There are truths about elementary arithmetic that are incomprehensible simply because there is neither space nor time enough for humans to even express them.

I’m not sure where exactly to gain purchase on answering this question. I can attempt some oblique passes on it based on material I remember.

Gödel thought his results in logic and proof theory implied dualism, the existence of God and that human consciousness transcended merely formal and mechanistic mathematics. He was the greatest logician of the 20th century, yet he believed his house appliances emitted poisonous vapors and starved to death when his wife was no longer able to prepare food for him. I’m pretty sure he ended up a Platonist (we explore mathematical reality rather than invent mathematical notions).

Roger Penrose put forth a “paradoxical, circular three-world picture” (my own wretched coinage just now) of reality. The material world appears to support the mental world which appears to support the platonic mathematical world which appears to support the material world. He also stands up for Platonism.

Many noteworthy mathematicians (Hilbert most famously) advocated formalism. You start of by setting the rules of the “game” (admitting axioms) then you play the “game” (deriving consequences from the axioms by the rules of logic). Hilbert however drew the line when it came to logic: he only considered classical mathematical logic valid. Today there are more logical systems then you can shake a stick at.

The Intuitionists hold that all valid mathematics is the result of human mental constructions, so they would answer your question wholly in the negative. Interestingly, the practices of the lead intuitionist mathematician (Brouwer) were formalized with what was widely considered by complete success (Heyting, Kolmogorov). This is ironic as Brouwer considered formalism to be “the enemy”. But, in turn, Gödel’s results show the limitations inherent to formalism (no system capable of supporting full arithmetic can be both complete and prove its own consistency).

Max Tegmark has some interesting ideas related to the general notions involved in your questions.

How would we know of a math that we couldn’t comprehend?
There are most undoubtedly elements of the universe we do not yet understand.

Both @ratboy and @The_Compassionate_Heretic have interesting points to make.

Consider the Classification of Finite Simple Groups. The full proof takes up volumes of text as the result of stitching together some 500 journal articles. Is it really possible to fully comprehend it all at once? How many have gone through the full proof with a fine-tooth comb? When is such a beast to be considered proved beyond a doubt? (It’s in instances like this were formal mathematics would have the advantage of allowing a computer to check the validity of a proof.)(This isn’t to say that someday, there might be a short and elegant proof.)

I had read somewhere (I think) that Wolfram’s Mathematica was being used to check formal proofs by constructing them in computable language. I think it might have been related to NKS, but I’m not sure (I’m rather wary of Wolfram’s craziness).

There are several functions that cannot be checked for validity, so I would agree that there are things intractable or non-computable given our current mathematics and computation abilities.

@Shuttle128 See also: Mizar and Metamath. This fellow also has tons of links.

To get back to what Zuma is asking, I think first you have to define what math is anyway, and on that there has not been universal agreement among humans, let alone humans and aliens (in our universe or otherwise)! ;)

@hiphiphopflipflapflop Very cool, I’m not surprised that that has been going on far longer than Wolfram gives credit. I’ll definitely look through some of that (though I’m not the greatest at formal logic so it seems a lot of it will be lost on me).

I’ve always been interested in what might have developed elsewhere under different circumstances. A good example of something that might have very different outcomes is the handling of imaginary numbers. How would aliens deal with the square root of negative one?

Do you know if mizar can compute new proofs based on proofs that already exist in the database, or does it only use proofs entered manually?

@Shuttle128 formalized mathematics allow for “existence proofs” that tell you that the non-existence of a given mathematical object leads to a contradiction. However these don’t offer any means of constructing said object. This is the crux of the split between the intuitionists and constructivists and the mainstream of pure mathematics. The former insist on an object being “constructable” or “computable” in principle. (But then what about practice?)

The great grandaddy of all formalization attempts is Russell and Whitehead’s Principia published in 1903 which famously took over three hundred pages before proving 1+1 = 2. :)

@Shuttle128 “I’m not surprised that that has been going on far longer than Wolfram gives credit.” Par for the course.

I’m not familar enough with how proofs are handled in Mizar to say definitely.

Anybody still following along might like to read a graphic novel that hit the New York Times’ bestseller lists recently.

@hiphiphopflipflapflop That is the best review I’ve ever seen for any book ever! Absolutely hilarious.

From what I’ve read (which is very little) it appears that in order to effectively use mathematics induction has to occur where we assume that formal proofs can be constructed of more than just the basics, and not the mathematical kind of induction that is a real proof. I would guess this is where the gap is between constructivists and those for practical application.

@Shuttle128 the real gap is more cultural I believe. All these matters are considered rather arcane by even most “working mathematicians” (as most “pure” mathematicians consider themselves to be as opposed to logicians and foundations people), let alone those who fall on the applied-side of the spectrum. (I suspect most of the latter are happily unware that they need to be “saved”.)

@Shuttle128 as an example, when I asked an instructor of mine about some of this, he said something along the lines of “I’m not interested in what the set theorists work on. As long as my universe has induction I can work in it.” and his lineage was pretty high power “pure” (a Havard Ph.D. of Tate who was a student of Emil Artin).

@hiphiphopflipflapflop: I’ve frequently read something to the effect that a working mathematician is a platonist during the week, and a formalist on Sundays.
The graphic novel Logicomix simplifies some of the issues involved; serious readers might wish to do some supplementary reading in the recently published reprint of Principia Mathematica, now available on Amazon for less than $20.00 per volume.

@ratboy Yes, I’ve heard that one too. For those who are confused, that means they imagine mathematical objects and work with them as if they actually exist and only when pressed by non-mathematicians about what they do (like at a dinner party) do they fall back on the “game” explanation if someone starts questioning the nature of their mathematical “reality”. Formalism just isn’t the way the vast majority of mathematics is practised, it’s sort of an idealization of it.

@ratboy Buying Russell and Whitehead!? Here’s where 1+1 = 2 is finally proved. Most pages in R&W are like that, symbolic manipulation. How many people went through the whole thing line by line?!

I recommend Martin Davis’ book for the lay reader. It turns out the intellectual battle between these points of view helped to create the foundations of computer science.

@hiphiphopflipflapflop:“How many people went through the whole thing line by line?”
I believe Quine claimed to have done so. That probably makes the total one.

I don’t really expect people to buy PM—I was amazed, however, at the price; until recently one had to shell out $300 – $1000 for a used set. Those who are curious can take a look online.

How would we recognize an incomprehensible universe if we ever came across one? Since, to state the obvious, we can’t comprehend what we can’t comprehend, we could not distinguish it from one where everything occurred for no apparent reason. Must there be laws? The probabilistic interpretation of quantum mechanics allows for some indeterminism. How much indetermism is possible?

The physicist George Gamow wrote a series of popular science books where the main character, Mr Tompkins, falls asleep at physics lectures and imagines a world where quantum mechanical uncertainty appears at a macroscopic level and the speed of light is much slower. Certainly the anthropic principle would prevent intelligent life in such a universe. Imagine what it would be like if objects regularly took random quantum leaps from place to place.

Even if there are laws, must they be consistent? Consistency is the one limitation on mathematics. You can make up any set of axioms, but it must not be possible to prove something as well as its contradiction, because the inconsistency will then work its way through the entire system. It has been suggested that physical constants have changed over time. What if they changed from place to place?

In short, metaphysically speaking, is God a mathematician?

I believe math is a human invention, not a discovery. I would assume that mathematicians only invent concepts they comprehend. Let’s look at this hypothetical scenario (inspired by Ray Kurzweil):

A benign superintelligence or technological singularity evolves (originally triggered by humans, but later modifying itself). We can assume that it will invent new mathematical concepts, maybe trying to explain dark energy or any other

http://en.wikipedia.org/wiki/Unsolved_problems_in_physics

Suppose it will show its grandparents (i.e. us humans) some of the new mathematical concepts. I would expect that some are not comprehensible to the human mind.

So the answer to your question is: yes

@hiphiphopflipflapflop Great answers. I am really intrigued by your reference to Max Tegmark and the mathematical universe hypothesis. That was exactly what I had in mind when I asked the question. (Damn, anticipated again.) What I found particularly interesting was the idea of the mathematician himself being a mathematical structure relating to the mathematical structure of the universe around him.

What kind of structure? Its got to be a fractal—the knower and the known forming a single recursive system. How would consciousness arise within such a system? It would seem to emerge when there is some kind of decision problem where one is being pulled this way or that but one does not have enough information to “decide” one way or another. Eventually, enough information accumulates and one “makes up one’s mind” and trips a parameter tipping point, causing the chaotic equilibrium to partially collapse or re-form in another state.

In this fractal which is based on Newton’s method for solving complex polynomials, there are five variables, each represented by a different color. As I understand it, there are an infinite number of solutions to this 5-variable problem. So, if you pick one pixel at random, the fractal tells you the value of the other four in relation to it. In this respect, the fractal represents all possible solutions. Imagine being able to assume a point of view inside one of these things. All the structure you see would coalesce (if that is the right term) around it. Somehow I can’t help thinking that all this has to do with superposition and how observation “decides” whether Schrodinger’s cat is alive or dead.

It also makes me wonder if consciousness can arise elsewhere in the fractal structure of the universe—say, at the edges of complexity just as tensions are about to be “resolved.” I wonder if a mountain experiences a kind of discomfort just before an avalanche, or does it require a population of individuals, such as bees in a hive, or cells in a brain. The journalist Tom Siegfried in his The Bit and the Pendulum describes what he calls The New Physics of Information, where all of physics can be re-cast in terms of information and computational processes.

I do not understand how anyone can say that mathematics is a human construct. For those who believe this, how does one explain that the laws of science can be described in mathematical terms? One would have to conclude that all of science is also a human construct. Then there is no need to do experiments, since it is all a construct of our imaginations. Does this not seem a bit absurd?

That’s the problem with only being able to detect the universe with your cheating, lying sensorium.

@Zuma You might also be interested in the John Wheeler’s speculation that information theory is the fundamental level of reality behind matter/energy and spacetime. He coined the slogan “It from Bit” for this. He also described the universe as a “self-excited circuit” using as a graphic a U with an eyeball (representing human observers) on top of the left end of the U. The eyeball is staring back at the right end of the U (representing the beginning of the universe and/or physics at the quantum level). He believed that to certain extent that we conjure ourselves and the whole of the universe and its history into existence by observations in the here-and-now.

@LostInParadise – It’s a philosophical question. There are pros and cons for invention versus discoveries. We might also see math as an obvious invention. Suppose there’s extraterrestrial intelligence somewhere. We can speculate that at some point they will construct a radio transmitter and receiver. Is it an invention? Yes. But inventions are based on physical reality. We discover photons. We discover they are both particles and waves. We use waves to create radio signals. We start counting stars. We invent (or discover) natural numbers. We divide sets of objects. We invent (or discover) prime numbers. And so forth.

matt, Consider a law like E=mc^2. It holds true with or without us. At the very least you would have to admit that numbers and numerical operations exist independently of humans.

@LostInParadise – Yes, E=mc^2 clearly is a discovery. Now take the Riemann zeta function which is of great significance in number theory. Discovery? Invention?

Matt, Are you saying that some of mathematics is discovered and some is created? It reminds me of the quote attributed to the constructivist mathematician Leopold Kronecker:
” God created the natural numbers; all the rest is the work of man.”

@LostInParadise – Actually, I think all of math is created. It helps us understand the physical world and its laws which are all discovered. E=mc^2 was discovered but a nuclear power plant was created.

Matt, Yes, but the nuclear power plant has to obey E=mc^2. We don’t have the luxury of making that up. And the equation involves real numbers. How can you say we invented real numbers when nature has been using them since way before we came on the scene? This is part of what Wigner refers to as the unreasonable effectiveness of mathematics.

@LostInParadise – It seems that nature doesn’t use real numbers after all. Only rational numbers (assuming Planck length and Planck time really is the limit). It makes our world discrete.

Okay, Matt, I will grant you that real numbers are a kind of fiction, but nevertheless a very useful one that allows equations to have infinite precision. It is only at the point where the results of the equaiton have to be applied to the real world that the real numbers have to be converted into rational approximations. And it still leaves open the question of why rational numbers can be applied so effectively to things in nature.

True. And the square root of -1 is useful too !

Matt, Actually it is just the irrationals that are fictitious. Of course a mathematician would say that the irrationals take up essentially the whole number line, since the rationals have measure zero. Pythagoras might have been relieved to know that those annoying irrationals that his group discovered/invented are not really needed and perhaps in some sense he was correct to view the universe as dancing to the vibrations of ratios of natural numbers.

Do you mean imaginary numbers? Real numbers include irrational numbers like e or pi.

No, I mean irrationals, the ones whose decimal expansion goes on forever without repetition. You said we only need rationals, so we can do away with the irrationals. The rationals are also real numbers. They just don’t take up any room on the real number line. They have measure zero.

We can express all the laws of science using rational numbers. In fact computers do not work with irrationals. Like I said, this still leaves open the question of why real numbers (in this case only the rationals) show up in the laws of science if they are just our invention.