As fans of The Hitchhiker’s Guide to the Galaxy recall, in the far future scientists set their massive computers to work to answer the question, What is the meaning of life? After a few million years, the computer had finished its work and delivered the answer: 42.

(Unfortunately, everyone had forgotten what the question was.)

Recently a team of mathematicians, tapping into idle time on a network of 500,000 computers, has plumbed the depths of this mystery. 42 can be written as the sum of three cubes!!!

As fans of The Hitchhiker’s Guide to the Galaxy recall, in the far future scientists set their massive computers to work to answer the question, What is the meaning of life? After a few million years, the computer had finished its work and delivered the answer: 42.

(Unfortunately, everyone had forgotten what the question was.)

Recently a team of mathematicians, tapping into idle time on a network of 500,000 computers, has plumbed the depths of this mystery. 42 can be written as the sum of three cubes!!!

I saw it. It has no practical implementation, but it's still cool.

I am currently obsessed with big numbers. After consuming Graham number now I am trying to understand TREE(3), which dwarfs Graham number.
Studying those numbers gives some understanding what infinity is. In the context of our natural laws those numbers are practically infinite, which in fact, they are not. They are to infinity as 1 is to infinity. Nothing. Yet, people keep talking about infinity as if it was just yet another big number. Like infinite number of universes, or laws of physics, or age of the multi-Cosmos.
Or "proton never decays". What is "never"? There is no "never".
Or the theological meaning of after-life. How can you live in after-life forever? A million years is plenty enough to get bored with whatever is out there, and million is really a tiny number. How to live Graham number of years? And it's still nothing. Or maybe the after-life is timeless. Suspended in some multi-dimensional realm and freed of any causality that gives sense of time. But then how do you enjoy it if there is no causality and your existence is just frozen.
Or being thrown into hell for infinity. How? Hopefully they give them a glass of water every Graham number of years. And yet, you end up drinking infinite amount of it. This shows what infinity really is. Or maybe infinity is an abstract concept and it simply does not exist in the natural laws.
I am a believer, but as a mathematician I struggle with understanding this theological meaning of "infinity".

That’s the entire concept of the Universe. Zero. If we assume gravity represents negative energy then the total sum of everything in the Universe is zero. The universe is created from nothing. We are all running on borrowed stuff. And one time it will be all returned and everything that exists now will become the original nothing. With no trace or memory.
It’s like splitting zero to plus one and minus one and giving them temporary existence. Until they collapse back to zero with no trace of previous existence.

Which brings another question: if an existence goes to a non-existence, including all memories and traces, then did it exist to begin with?
If someone draws a picture on sand and then the moment later she forgets the picture, and comes a wave and washes the picture, then did the picture exist at all?

That’s the entire concept of the Universe. Zero. If we assume gravity represents negative energy then the total sum of everything in the Universe is zero. The universe is created from nothing. We are all running on borrowed stuff. And one time it will be all returned and everything that exists now will become the original nothing. With no trace or memory.
It’s like splitting zero to plus one and minus one and giving them temporary existence. Until they collapse back to zero with no trace of previous existence.

Which brings another question: if an existence goes to a non-existence, including all memories and traces, then did it exist to begin with?
If someone draws a picture on sand and then the moment later she forgets the picture, and comes a wave and washes the picture, then did the picture exist at all?

When I saw this thread, I immediately thought of the movie, "42." When Jackie Robinson integrated the major leagues in baseball. I find it relevant, in a quirky way, to a discussion about infinity, mathematical beauty, and the meaning of life. My favorite quotes from the movie, a sublime one and a tongue-in-cheek one:

Jackie Robinson: You want a player who doesn't have the guts to fight back?

Branch Rickey: No. No. I want a player who's got the guts not to fight back.

It's such a powerful idea; and one that goes against so much of conventional wisdom.

Branch Rickey: [Referring to Jackie Robinson] He's a Methodist, I'm a Methodist... And God's a Methodist; We can't go wrong.

Studying those numbers gives some understanding what infinity is. In the context of our natural laws those numbers are practically infinite, which in fact, they are not. They are to infinity as 1 is to infinity.

Mathematics is a language that allows us to express common patterns in different fields of science and technology. There is a concept of infinity, and you can intuitively imagine it as some super big value, but in the end it's a separate abstract concept with its own properties.

And you are not even obliged to speak the same language. As a programmer, I often deal with IEEE 754, in which infinity is just another number with some special properties.

But science doesn't work the same way. If you observe some physical law in a 10[SUP]26[/SUP] meters radius around yourself, you still can't be sure this law will apply in 10[SUP]30[/SUP] meters from you. Who knows, maybe talking about such distances will make as little sense to a 22nd-century astrophysicist as if an ancient Greek told us about the mechanical properties of the firmament.

Hey, what if you need three cubic water tanks of integer sizes and the total volume of 42? You'd need to get cubes with the size of 12602123297335631, 80435758145817515 and -80538738812075974.

What I am saying is that physics deals with the reality. And the assumption is that reality does not reach to the true mathematical infinity. Everything in reality is discrete and finite. It can be huge, but still finite. Including all possible scenarios of all possible universes with all possible natural laws and all possible initial conditions. Even if that number goes to 10^10^10^10^10^10^10^10, which it doesn't, but even if we imagine it does, then it's still finite.
If you take Graham's number or TREE(3), they dwarf everything that can be described in the language of reality. Yet, they are still finite. But because they dwarf the reality then we can assume that TREE(3) for the reality is practically infinite. That means is impossible ever to reach. Even in theoretical calculations. It's as good as the true infinity.

As to math as a language, there are several views on that. Some say, and I personally believe in it, that it's not a language of reality, but it's actually the only objective reality itself. The reality is the realization of math, rather than math being description of reality. The other way around. There is even a theory of it by Max Tegmark and some other theoreticians. That math is actually the only reality. In fact, math does not need any outside reality. Math does not even require a mathematician. A parabola is an objective existence and it doesn't need a mathematician to draw a formula for it. It is a solution of an objectively existing formula. It's timeless, it has no causality or a need of creation. And the universe is nothing more than a realization of objective formulas combined with probability laws. And it's based on philosophical assumption that everything that can exist will exist.

Hey, what if you need three cubic water tanks of integer sizes and the total volume of 42? You'd need to get cubes with the size of 12602123297335631, 80435758145817515 and -80538738812075974.

As to math as a language, there are several views on that. Some say, and I personally believe in it, that it's not a language of reality, but it's actually the only objective reality itself. The reality is the realization of math, rather than math being description of reality.

Then we don't know what math it is. For millennia we believed that we live in a Euclidean space, but in the 20th century we learned to see the world as some superset of Riemannian spacetime and Hilbert space of quantum states. This doesn't mean that Euclidean geometry disappeared or has never existed, it only means that the language we used doesn't meet our new needs.

I think I would say rather that we don't know what mathematics is at all. (I don't know what math is, but I know it when I see it. )

For instance, consider the concept of "number." If numbers are such a big deal, what are they? What is the number two (never mind it's ontological status)?

Suppose we try to teach a child what "two" means. We tackle this task by showing him two ducks and then showing him two kittens and then showing him two stars, and saying "one, two" each time. Somehow or other by repetitions of this game we hope that the child starts to acquire some intuitive feel for what it means to have two particular things. But does that address the question of what the number two actually is? This pair of ducks is not, after all, the Number Two -- or any number. It is a set of ducks.

And yet, in the end, that is all we can do. A plausible definition (in the mathematical sense) might be, the number two is the class of all sets that can be put into a one-to-one correspondence with the set {{}, {{}}}, or something like that. Thus the set of two ducks is a representative of the Number Two in all it's profound and abstract glory.

But in fact all this cogitation is doomed to fail. The most important mathematical discovery of the twentieth century is embodied in the works of Kurt Gödel in the 1930s. Gödel took on the problem of whether it is possible to put mathematics on a sound logical foundation, and he “proved mathematically” that it is not.

Gödel’s Incompleteness Theorem (1931) says that in any attempt to reduce mathematics to pure logic (provided that the system is sufficiently robust to allow sets to be the subjects of quantified sentences within the grammar of the system) there will always be theorems that are true within the system but cannot be proven true within the system. This is quite distressing, because that is exactly what mathematicians do – they try to prove that true theorems are true.

Thank goodness Gödel’s theorem itself did not turn out to be one of those unprovable true theorems. If it had, we would never have discovered that our millennia old quest to set mathematics on a secure foundation of logic has already failed before we begin.

Gödel’s Second Incompleteness Theorem was an even more serious blow to our one-time hopes. If an axiomatic system is consistent, then it cannot be proven to be consistent. But if an axiomatic system is inconsistent, then it can be proven to be consistent (and hence it is inconsistent, being both consistent and inconsistent).

So if we are ever able to prove that our system of fundamental mathematical axioms does not contradict itself, then we can be certain that in fact it does.

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Kurt Gödel immigrated to the United States form Austria in 1940, fleeing the Nazis. He was afraid to attempt an escape westward across the Atlantic, so he hopped on the trans-Siberian railroad and went the other way, eventually arriving in Japan. From there he took a boat for San Francisco and then a train across the U.S. to Princeton, New Jersey, where was befriended by Albert Einstein. In due course he applied for U.S. citizenship, and Einstein took him down to the judge’s chambers to be interviewed for citizenship.

Gödel, having carefully studied the Constitution of the United States in preparation, went on a big rant in front of the judge about a horrible flaw that he had discovered in the U.S. Constitution that would allow an unprincipled president to declare himself a dictator, and there would be nothing that Congress, the Courts, the States, or the People could do about it.

Einstein was able to calm him down and Gödel did become a U.S. citizen.

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In common with many prominent mathematicians who specialize in set theory, logic, and the foundations of mathematics, Gödel eventually lost his mind utterly, succumbing to paranoid delusions. He died of starvation in the hospital when he refused to eat, believing that the doctors were trying to poison him.

Eminent mathematicians who study the logical foundations of mathematics often become reclusive and odd in later life. This invites us to wonder if the study of mathematical logic can drive a person crazy, or conversely, if people who are predisposed to mental illness are attracted to this field of study.

This is why we differ from other mammals. We think abstract. The number three is not an estimate of too many enemies or too little to eat. It’s a class of things that have three elements. Three houses are equivalent to three books and to three days. Or to simply three of nothing. It’s the same thing.

I’m wondering if the Universe, and the laws of nature, does no fall into the incompleteness theorem. It cannot be understood, because it cannot be understood from within inside. A creature born from it does not have a view or foundations to formulate its definitions or logic. Not because we are not intelligent enough, but because it’s logically impossible.

My biggest disappointment from the incompleteness theorem is logical inability to find a set between naturals and reals. Intuitively it should be obvious: all subsets in 2^c are almost visible. How can we not know?
Is there mathematics where existence, or non existence, of such a set can be proven?
But if there is, isn’t a Godly mathematics, where things are because they objectively are? Subsets of R appear to be such objective entities. It’s either this cardinality, or the other. Doesn’t God know the answer?
Before Godel it was like discovering it. We just to have to keep looking. And now such a disappointment.
It’s like Fermat theorem. It’s objective if it’s true or false. It’s not axioms and language. Just run all the numbers and check. We know now after it’s finally proven, but even if it were not, it’s simply an objective existence.
Subsets of R should be the same thing. Just run all possible subsets and “verify”...

Let’s get back to Fermat. It turned out to be provable, it took 300 years, so it was not trivial, but let’s make a mind experiment. Let’s say it turned out to be unprovable. It’s a statement that mathematics cannot prove that it’s true. So we don’t really know if it’s true. It could be false, meanings there are some numbers a^n+b^n=c^n.
So we have a statement that we don’t know if it’s true and we also know mathematics with its axioms can not prove or disprove it. So the only solution is to somehow guess the combination that disproves it or count to infinity and this way prove it. By brute force, not by mathematical language. Because the existence or non existence of such numbers is a cold objective fact. There are such numbers or not. God knows it.
The same with the set between naturals or reals. It exists or not. It’s a cold objective statement that cannot be proven or constructed with mathematical language, but can be “proven” by manually and forcefully verifying every subset of 2^c. Of course it’s a ridiculous task, but God can do it. That means that subset exists or not.

Eminent mathematicians who study the logical foundations of mathematics often become reclusive and odd in later life. This invites us to wonder if the study of mathematical logic can drive a person crazy, or conversely, if people who are predisposed to mental illness are attracted to this field of study.

As a mathy person (not exactly a mathematician, but depending on your definition, not exactly not a mathematician) and also a devout Christian, I just say that mathematics is God's instruction manual for the universe that perhaps humans don't even have the right to fully understand and thus don't bother to ponder the inner workings of its logical foundations too deeply. I think that's a nice, healthy middle ground. I can get myself pretty freaked out if I think about it too long.

That being said, there are some things that I do think I understand (like the concept of numbers mentioned earlier), but it's more of a feeling or a physical sensation than anything I could ever put into words. Maybe that's how God keeps His instruction manual secret: lets some people understand it (at least at a very rudimentary level) but not verbalize it, and others verbalize and share information but not quite be able to piece it together.

In my opinion, the only criterion of objective truth is the ability to predict observations in different conditions. No physical theory can reach the strict 100% precision and reliability, because there is always a chance that some experiment will expose an exception from the rule. Euclidean geometry (as a basis for physics) was close to 100%, but at some point it turned out to be only accurate enough for our usual conditions.

In maths, things can be proven with the strict 100% reliability, but they will only stay proven within the axiomatic system you used. So far, integral arithmetics has been working fine for every observed countable thing in our universe, but some day we may discover sepulkas for which 2+2 randomly makes from 3 to 7.

I'm not saying that mathematical patterns are just a random coincidence, but I see no reason to believe that there is some deeper truth behind them than in our material world.

My biggest disappointment from the incompleteness theorem is logical inability to find a set between naturals and reals. Intuitively it should be obvious: all subsets in 2^c are almost visible.

C = continuum. Like all real numbers. Cantor proved it’s larger than naturals, meaning it’s not countable.
2^c = All subsets of continuum. Continuum hypothesis says that some of them might be not countable, but still not equal to the continuum.
Visibility of a set = it’s my metaphor. You can look at the continuous line and almost like see all the subsets. They are right before your eyes. Kind of. Metaphorically.

In my opinion, the only criterion of objective truth is the ability to predict observations in different conditions.

In my opinion objective truth is whatever is happening regardless of us observing, describing or understanding. It’s like Fermat theorem. It’s objective if none or some numbers satisfy a^n+b^n=c^n. We may prove it or not, but it’s objective. It’s out there in a very defined concrete state.
The answer doesn’t need us trying to find it.

The same is what I understand about continuum hypothesis. It’s out there. Truth or not. Our axioms are too short to reach there, but I know it must be there already determined.
The answer is done. We just can’t get it.

Or maybe not. Maybe I’m wrong. Maybe the answer DOES depend on axioms. Maybe with some axioms there is a set between N and C, but with others there isn’t ?

Or maybe not. Maybe I’m wrong. Maybe the answer DOES depend on axioms. Maybe with some axioms there is a set between N and C, but with others there isn’t?

This is actually correct, speaking strictly in terms of mathematical logic. In fact, in my opinion that is the best way to interpret the "independence of the continuum hypothesis."

What Goedel actually proved (1940) was, the continuum hypothesis cannot be proved true within standard models of set theory and foundations of mathematics. In 1963, Paul Cohen (a rare great set theorist who did not go crazy ) proved that, conversely, the continuum hypothesis cannot be proved false either.

A more precise way of saying it is this: If some version of standard axiomic set theory is self-consistent, and if you throw in the Continuum Hypothesis as an additional axiom, then the theory is still self-consistent. And if you throw in, instead, the denial of the Continuum Hypothesis as an additional axiom, then again the new expanded system is still self-consistent.

You pays your money and you takes your choice.

So in that sense, yes, there are some systems of logic in which the continuum hypothesis is true and other systems of logic in which the continuum hypothesis is false.

(Standard axiomatic approaches contain such axioms as, "At least one set exists." This is a great comfort to those mathematicians who devote their careers to studying them. )

In particular, the idea that we can "almost see" all the subsets of the real line, or that, if there is an oddball set with cardinality strictly between that of the integers and that of the real numbers then it must be right there in our face waiting to be discovered -- I think the situation is murkier than that.

The reason that these paradoxes involving infinite sets are so perplexing, I believe, is that all of our intuition about "the properties we think that sets ought to have" are derived from our experiences with finite sets. IMO we just do not have any reliable intuition or "common sense" about what we might or might not find in the mathematical universe of infinite sets.

I agree that this is the very definition of "science" and the scientific method.

I would use this language. What mathematical physicists do is create (mathematical) models of reality. General relativity, for instance is a system of differential equations that models gravitational phenomena. So what distinguishes a good model from a bad one?

Well, besides being internally consistent and accounting for all the observations that we have made so far, a successful mathematical model will suggest novel experiments (that is, new experiments that have never been thought of before) and accurately predict their outcomes.

I'm not saying that mathematical patterns are just a random coincidence ...

To me, that is the huge question in all of science. How is it possible that this model building works at all? After all, a mathematical equation is just chalk marks on a blackboard. How is it possible that the mighty stars and planets in their courses care what we scribble on our blackboards?

More precisely, we seek a set X whose cardinality is between the cardinality of the natural numbers and the (strictly bigger) cardinality of the real numbers.

Cardinality is supposed to mean, vaguely, "how many members does a set have." The cardinality of the set {p,q,r} is 3. So far so good. Unfortunately, when we pass to infinite sets, this seemingly elementary notion "how many" becomes a vast and storm-tossed ocean. We can no longer rely on common sense and common experience as guides, and we must confront impossible logical paradoxes that seem better designed to devour us than to guide us to the truth.