42, the meaning of life

dante

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Continuum hypothesis says that some of them might be not countable, but still not equal to the continuum.
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.


Ah, I vaguely remember our lecturer reordering rational numbers, mapping them to ordinals. I see what you mean.

In my opinion objective truth is whatever is happening regardless of us observing, describing or understanding.

Let's sort out our terminology.
- The objective reality is what happens outside of our consciousness (i.e. everything except qualia), regardless of whether we observe it.
- The objective truth is a simplified model of the objective reality that reflects important (for us) features with an adequate (for us) accuracy.
- The criteria of objective truth is how we decide how close some hypothesis/theory is to the objective truth.

All the mathematics comes down to statements like "If [axioms] then [theorem]." It doesn't automatically make the axioms the objective truth, thus the theorems and the whole maths can't be called objective either.

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 ?

The answer always depends on the axioms. It's enough to assume that ∀x, x≠x, then any equation will be false, and Fermat's too. I believe Peano axioms will remain consistent if we redefine this one, even though the whole system would be practically useless (not that mathematicians worry about such minor issues :biggrin:).

besides being internally consistent

Internal consistency is an important point I missed.

So OK, we have mathematical patterns with internal consistency (which allows us discarding obviously implausible models). We have a mathematical apparatus that allows us bringing those models to a common basis.

So, what do you mean when saying that the material world is probably a result of mathematics? From a naive idealist point of view, an atom and a star both implement some common idea, but from the scientific point of view, you only need to describe an atom, and the star will be a necessary result of atom's properties.

Today's most fundamental area of physics is called quantum field theory, and most of the observable universe on most levels is a result of it. Is there really a place for Platon's idea?

That's something I never understood about God. (I understand everything about God except this one thing. :) )
Why does he want to keep his plans secret?
Lazy programmers never document their code, making their successors suffer. :)
 

Manitou

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So, what do you mean when saying that the material world is probably a result of mathematics? From a naive idealist point of view, an atom and a star both implement some common idea, but from the scientific point of view, you only need to describe an atom, and the star will be a necessary result of atom's properties.

I’m not a physicist, but I love reading popular science literature, so I understand there are no concrete atoms in the universe. Like sand pebbles flying in space. In the sense there is no concrete substance in the universe at all. If you keep zooming in into the microspace you will never find anything. On the microscopic quantum scale there are only probabilistic functions, which on macro scale combine together to form fields, bindings and illusion of matter.
Some people even think there is no space, but on micro scale the universe is simply a graph of nodes and edges. And there is no sense of distance between nodes. If two nodes are connected via an edge then it makes no sense to say if that edge is one Planck length, or is 10 billion light years. It’s just one graph edge. And there is no concept of space where those nodes are embedded. They are embedded in nothing. They do not float in space. They define the space. They are the space.
 

dante

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On the microscopic quantum scale there are only probabilistic functions
They are probabilistic waves that can be described with a function. A wave is a physical entity, while a function is a mathematical entity.

And to make it clear, probabillity doesn't mean that a particle may or may not exist in some place. It means that the chance of interaction of two waves in this place is proportional to the product of their functions (it involves an integral of Ψ₁*Ψ₂ over (x,y,z,t)).

I understand there are no concrete atoms in the universe. Like sand pebbles flying in space. In the sense there is no concrete substance in the universe at all.

You must be talking about identity of particles. It means you cannot mark an electron and later recognize it among other electons. Moreover, the waves of several electrons interact as if they were a single big wave (until we reach superconductivity). But these, again, are physical properties that were mostly discovered empirically.

Some people even think there is no space, but on micro scale the universe is simply a graph of nodes and edges.

This hypothesis doesn't simplify our model of the Universe and hasn't predicted any new phenomena so far. In other words, it's just someone's imagination.
 

Manitou

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This hypothesis doesn't simplify our model of the Universe and hasn't predicted any new phenomena so far. In other words, it's just someone's imagination.

The point is that limiting ourselves to physical reality will never answer this question: why is reality to begin with? What does it come from? Why is anything?
You may give an answer that in some unknown quantum realm a fluctuation happened which spawned the Big Bang. The thing is even if that is true, it doesn’t answer where that quantum realm came from? The primordial fluctuation must have happened in some context. Or according to you, in some physical context. But that context also has to have its own context. And it keeps going back.
But if you keep going back in the causality chain you will never find the very first domino that triggered the beginning of everything.
The only explanation to those never ending causality dominoes is pure math. Pure math is the only concept we know that doesn’t need a cause. It doesn’t need a creator and it doesn’t need any outside context. It’s the only truly objective absolute.
You will say math is just a language of axioms and logical rules. Exactly! Some combination of axioms and rules creates a context in which our realm is possible. And because it is possible then it is born.
Some other axioms and logics create other contexts which do not favor worlds like ours. So they are not born.
But whenever anything is possible, in whatever shape and laws, it gets created. Simply because it’s possible.
I’m not sure if that’s what Max Tedmark is suggesting, maybe I’m going too far, but I cannot find a better explanation for the endless causality problem than pure math.

They are probabilistic waves that can be described with a function. A wave is a physical entity, while a function is a mathematical entity.
Which to me is equivalent. What comes first: chicken or egg? The wave or the function? Is the function a description of the wave, or the wave a realization of the function?
 

dante

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Ancient philosophers belived that our Universe was born by Chaos. The scholastics assumed that it was created by God the Father. Some modern philosophers claim that it may be a result of the mathematics. But I think they still rush too much. In my opinion, such modern hypotheses are as baseless and naive as belief in pagan gods.

Moreover, they don't answer the original question, why there are mathematical and logical laws in the first place. Is there anything more fundamental than them? Why is there anything at all?

Man's quest for knowledge is an expanding series whose limit is infinity, but philosophy seeks to attain that limit at one blow, by a short circuit providing the certainty of complete and inalterable truth. Science meanwhile advances at its gradual pace, often slowing to a crawl, and for periods it even walks in place, but eventually it reaches the various ultimate trenches dug by philosophical thought, and, quite heedless of the fact that it is not supposed to be able to cross final barriers to the intellect, goes right on.
your compatriot

Do you want to really see the likely limitations of physics? Have a look at the hard problem of consciousness.


What comes first: chicken or egg?
Dinosaur eggs existed long before chickens. :)
 

Mathman

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The only explanation to those never ending causality dominoes is pure math.

As much as like mathematics, I think that this conclusion is hard to establish either by logic or by evidence. I think that the question, "Why is there something instead of nothing" cannot be investigated by the mathematical method (arguing from premises to conclusions by following generally accepted rules of logic), nor by the scientific method (forming hypotheses and testing their predictions by experiment).

I think the answer to the question is "I don't know." For some reason, human beings are incapable of saying "I don't know" when confronted by a puzzling riddle. Instead they will make up one answer after another (sequentially reasoning, "but what else could it possibly be?"), and then go to war over who's answer is right. ;)

To go back for a moment to the Continuum Hypothesis and such theorems,, the brilliant logician Georg Cantor figured it all out. He invented a mathematical definition of what it means for one infinite set to be bigger than another infinite set. Then he and his successors proved a lot of theorems along the lines of, "IF you accept my definition of 'more than,', THEN there are more points on the real line than there are integers."

That is all mathematics ever does or pretends to do: prove theorems of the form IF p, THEN q.

Was Cantor's definition right? Definitions cannot be right or wrong -- they are just general agreements about how we will use language, in this case mathematical language.

Cantor was persecuted all his life by the mathematical establishment of Germany, led by Leopold Kroniker. Kroniker felt that Cantor's work with infinite numbers was blasphemous, because only God (and not us) can comprehend infinity. He accused Cantor of proving his theorems not by mathematics but by witchcraft, thus undermining the morality of ernest young seekers-after-the-truth.

Cantor spent his last years in an insane asylum, obsessed with proving that the true father of Jesus was Joseph of Arimathea and that Shakespeare did not write the plays attributed to him.
 

Mathman

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Today's most fundamental area of physics is called quantum field theory, and most of the observable universe on most levels is a result of it. Is there really a place for Platon's idea?

To me, what is most telling is that we refer to the whole edifice of quantum theory (so far) as "the standard model." I think that practicing physicists should take that to heart. What physicists do is build models of reality. But the model is not reality, any more than a model airplane carved out of balsa wood -- however beautiful and useful -- is not the actual airplane that it models. To say that certain probabilities are modeled by an integral (a mathematical concept we invented only a few centuries ago) -- well, that's cool and hooray for us, but I think we should be modest about our claims of understanding the actual world.

As for Plato and Aristotle, from the modern perspective every single thing that they ever said about science turned out to be wrong. Still ... their ideas dominated Western thought for 1500 years. What will be the fate of our current theories?

The greatest scientific model of all time was Newtonian physics. And this model is doubly to be praised because the scientists of the seventeenth century (Galileo and Newton in the fore) not only invented a spectacularly successful model for almost all physical phenomena that had been observed up to that point, but also they invented the criteria (the scientific method) by which such models are to be judged.

A couple of decades ago String Theory was the big deal. But it petered out without accomplishing anything and now is more or less just an embarrassment to physics. (As they say, "String Theory is not even wrong.") :(
 

Manitou

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I think the answer to the question is "I don't know." For some reason, human beings are incapable of saying "I don't know" when confronted by a puzzling riddle. Instead they will make up one answer after another (sequentially reasoning, "but what else could it possibly be?"), and then go to war over who's answer is right. ;)

Obviously, if I really knew and could prove it then I would receive a Nobel :D. Of course, we don't know. The thing is it's great that we don't stop at "I don't know", but we keep looking and guessing. And even saying "Gods did it" is much better than "I don't know". :)
And coming back to the thing we are talking about: from all the concepts and models and actually everything we ever created and observed, mathematics is the ONLY ONE that doesn't need a cause. Everything else does. Mathematics exists above cause and is completely independent on it. Does not need a context, time, space, a creator or an observer. Two falses are always a false and two trues are always a true. In all possible reasonable logical systems induction always stands. And those are true regardless if there is anything to use, observe it or contain it. Even if there are no universes at all, even in a complete cold nothingness, the laws of induction still stand.
 

Mathman

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And even saying "Gods did it" is much better than "I don't know". :)

This is the part that I do not agree with. I think it is better to say truthfully, "I don't know," than to say falsely, "The great god Thor did it."

Imagine ancient people who see lightning and ask themselves, "What the heck is that?" Surely it is better to say, "I don't know," than to say, "That's Thor up on his mountaintop striking his mighty anvil with his magic hammer."

If we say Thor did it, that's the end of our investigation. If we say, I don't know, that's what spurs us onward.
 

Manitou

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This is the part that I do not agree with. I think it is better to say truthfully, "I don't know," than to say falsely, "The great god Thor did it."

Imagine ancient people who see lightning and ask themselves, "What the heck is that?" Surely it is better to say, "I don't know," than to say, "That's Thor up on his mountaintop striking his mighty anvil with his magic hammer."

If we say Thor did it, that's the end of our investigation. If we say, I don't know, that's what spurs us onward.

Agree or not, but religion has been the biggest civilization drive since we consumed Neanderthals. It was the biggest unifying factor first for tribes, then societies and then for civilizations. Nothing unifies people more than common beliefs and common faith. It led to biggest destructions and to biggest creations. Nothing gives people more energy and creativity than fanaticism under one god. :)
And excuse me, the biggest art pieces were created driven by religions. :)
 

el henry

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This is the part that I do not agree with. I think it is better to say truthfully, "I don't know," than to say falsely, "The great god Thor did it."

Imagine ancient people who see lightning and ask themselves, "What the heck is that?" Surely it is better to say, "I don't know," than to say, "That's Thor up on his mountaintop striking his mighty anvil with his magic hammer."

If we say Thor did it, that's the end of our investigation. If we say, I don't know, that's what spurs us onward.

Finally something I know a little bit about (having been learning much by reading this thread:agree:)

I'm not saying you think this, but just to continue the discussion, the ancients didn't really have the same concept of "religion" as popularly thought today. (In fact, Romans didn't have a common definition of religio). So it was perfectly possible and acceptable and OK to both say "Zeus is throwing thunderbolts across the sky" and "I wonder what those streaks are and what causes it". No-one in the ancient Western world would think, as a matter of religious "belief", that "Zeus is throwing thunderbolts across the sky" ended the discussion. Thus, so many Ancient Greek scientists and mathematicians:thumbsup: Romans, well, maybe some.....;)

That narrow concept of that religious belief requires adherence to all sorts of other "facts" did not come until much later, and of course, isn't applicable to every religion now either. My former rector was ABD in some kind of astrophysics and an instructor at Local Large Private University. He wore his collar on purpose to classes so that students would see if was possible to be a person of faith and a questioner at the same time.

I don't know if it worked, but I hope it did :)
 

Ded Moroz

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As for Plato and Aristotle, from the modern perspective every single thing that they ever said about science turned out to be wrong. :(

You are right about Aristotle.
You are also right about Plato, if you read him in prose.
 

Mathman

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Nothing gives people more energy and creativity than fanaticism under one god. :)

And excuse me, the biggest art pieces were created driven by religions. :)

I am all for fanaticism and art (especially art). :rock:

The point I was trying to make is, give unto Caesar what is Caesar's and unto God what is God's. These big questions ("Why is there something instead of nothing") cannot be answered by applications of the scientific method nor of the mathematical method (two praiseworthy human inventions, to be sure).

I am also, in these investigations, a big fan of telling the truth. If someone asks me a question and I do not know the answer, then "I don't know" is God's honest truth. :yes:
 

Mathman

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*whole post*

:clap: :clap: :clap:

He wore his collar on purpose to classes so that students would see if was possible to be a person of faith and a questioner at the same time.

I am far from well-versed in ancient Greek religion and philosophy. (Although I am a big fan of the inscription greeting attendees at Plato's Academy: "Let no one ignorant of geometry enter here." :) ) The original formulation of the Big Bang Model, the expansion of the universe and what later became known as Hublle's Law, was a Catholic priest (George LeMaitre).

As for modern times, I think it is a common stance to say that there is no essential conflict between science and religion because they speak about different aspects of the human experience.

Still, I think there is a huge difference between the approaches of modern Western faith-based religions and the scientific method. Namely, in faith-based religions it is a virtue to believe things that that cannot be tested or proved. In science, this is the cardinal sin. The heart of the scientific method is not faith but its opposite, skepticism. I won't believe it until you show me -- and even then, I won't believe it and I will do my darnedest to show that the claim is false.
 

dante

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A couple of decades ago String Theory was the big deal. But it petered out without accomplishing anything and now is more or less just an embarrassment to physics. (As they say, "String Theory is not even wrong.") :(

The problem is in the word "theory". Call it "string hypothesis", and everything will fall into place.

As for Newtonian mechanics, quantum mechanics and special relativity, even though they don't claim to explain everything observable, they have their convenient scopes of scale and tasks. They remain true, as long as we use them in certain limits and deliberately neglect the expected deviations.

Imagine ancient people who see lightning and ask themselves, "What the heck is that?" Surely it is better to say, "I don't know," than to say, "That's Thor up on his mountaintop striking his mighty anvil with his magic hammer."

Sure. Proselytism among ancient people can be very dangerous.

These big questions ("Why is there something instead of nothing") cannot be answered by applications of the scientific method nor of the mathematical method

After all the claims by Plato and Aristotle that proved to be wrong, I wouldn't rush with such statements.

"If an elderly but distinguished scientist says that something is possible, he is almost certainly right; but if he says that it is impossible, he is very probably wrong." (but that's not for sure. :))

Agree or not, but religion has been the biggest civilization drive since we consumed Neanderthals. It was the biggest unifying factor first for tribes, then societies and then for civilizations. Nothing unifies people more than common beliefs and common faith. It led to biggest destructions and to biggest creations. Nothing gives people more energy and creativity than fanaticism under one god. :)
And excuse me, the biggest art pieces were created driven by religions. :)

Economy. It was economy that allowed nations and ideologies to raise and spread. Among thousands of cults those became world religions that best served the interests of economy. Nothing unifies people better then the need to exchange vital goods; and once the need passes, the state falls apart in a matter of decades. Concentration of capital and productive forces has always been the main factor in creation of the most marvelous wonders of human civilization, including art.

But that's not for sure either. :)
 

Mathman

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The problem is in the word "theory". Call it "string hypothesis", and everything will fall into place.

Not entirely, in my opinion. I believe that scientists use the term "scientific hypothesis" in a way that is special to their discipline, as do mathematicians in their use of the language "mathematical hypothesis." We certainly don't mean vague speculation or flights of fancy.

To me, a "scientific hypothesis" must contain (perhaps hidden) instructions for experimental tests which in principle possess the ability to falsify the hypothesis if the experiment doesn't work out as predicted.

In my opinion, this is where the critics of the String Hypothesis have the upper hand over their colleagues. String Theory is such an amorphous monster that it can "account for" virtually any result that might occur for any experiment. It has become, in this regard, like a sounding brass or a clanging cymbal, full of sound and fury, signifying nothing.

In mathematics, a "hypothesis" means something different, I believe. It is a statement that we are willing to posit (we don't really care if it is true or false) just to see what logical conclusions it leads to. In other words, mathematicians are not concerned with the truth of the conclusions per se; their interest is in the soundness of the implication.

Now about whether either science or mathematics is equipped to deal with the question, "Why is their something instead of Nothing" ... :)

Thousands of years ago children figured out how to troll adults. Whatever the adult says, the child just responds, "Why?" The child will always win.

Suppose we say, the reason that there is Something instead of Nothing is because of God. Well, why is there a God instead of no God?

The reason there is Something instead of Nothing is because any quantum system will seek it's lowest energy state, and the energy state of the vacuum is not the lowest possible. Why is there such a law of nature instead of there being no such law?

There is Something instead of Nothing because in the infinite dimensional multiverse of possibilities everything that can happen does happen somehow or somewhere.

Why is there such a metaphysical principle instead of there being no such metaphysical principle. (Or for that matter, why does the infinite multiverse of possibilities exist instead of there being no such infinite multiverse of possibilities?)

Turtles all the way down. ;)
 
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Mathman

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To Manitou:

I thought of how to explain the distinction between problems like the independence of the continuum hypothesis and problems like Fermat's Last Theorem.

All mathematical entities (definitions, rules of logical inference, proofs of theorems, etc.) consist of finite strings of symbols. This is why it is possible for the magical set that might be the contradiction to the continuum hypothesis to be sitting right under our very noses, but we will never able able to "see" it within the rules of mathematics. We would not expect that every one of the 2[sup]c[/sup] uncountably infinite subsets of the real numbers can be completely described by any finite string of symbols.

For Fermat's Theorem, though, suppose that there is a counterexample. This would mean three finite integers that have a certain arithmetic property. These three numbers can certainly be completely specified by finitely many symbols (for instance, you could write them down in decimal notation). Likewise, it requires only finitely many elementary computations to verify that they have the property that we claim.

These ideas were made precise largely by the work of Alan Turing in the late 1940s. Turing pioneered the question of what sort of problems can, in theory, be solved by a computer that just chugs along flipping finitely many logic gates for a finite time.
 

Manitou

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To Manitou:

I thought of how to explain the distinction between problems like the independence of the continuum hypothesis and problems like Fermat's Last Theorem.

All mathematical entities (definitions, rules of logical inference, proofs of theorems, etc.) consist of finite strings of symbols. This is why it is possible for the magical set that might be the contradiction to the continuum hypothesis to be sitting right under our very noses, but we will never able able to "see" it within the rules of mathematics. We would not expect that every one of the 2[sup]c[/sup] uncountably infinite subsets of the real numbers can be completely described by any finite string of symbols.

For Fermat's Theorem, though, suppose that there is a counterexample. This would mean three finite integers that have a certain arithmetic property. These three numbers can certainly be completely specified by finitely many symbols (for instance, you could write them down in decimal notation). Likewise, it requires only finitely many elementary computations to verify that they have the property that we claim.

These ideas were made precise largely by the work of Alan Turing in the late 1940s. Turing pioneered the question of what sort of problems can, in theory, be solved by a computer that just chugs along flipping finitely many logic gates for a finite time.

That's what I am saying. For each subset A in 2[sup]c[/sup], we (or the gods) examine all functions f:A->C (C[sup]A[/sup] of them) to see if they are bijections between A and either N or C. And the answer is already predetermined, with ZFC or not. If we find a subset that none of such function satisfies it then we found a subset that fails CH. However, if for each subset there IS such function somewhere in C[sup]A[/sup] then CH stands.
The answer to CH is right there. IT only tells us that we will never find it. God knows it. Assuming God knows how to count C[sup]C[/sup] times. It might be too much even for God.

The same applies to Fermat. We simply run checks on each quadruple (a,b,c,n) in N[sup]4[/sup]. If we run across a vector that satisfies the equation then Fermat fails. Again - it's already predetermined. It could be immediately disproved if some supercomputer runs across such numbers. Some kind of computational mathematics. Must be very uncomfortable to theoretical mathematicians. But God already knows if such numbers exist. And God doesn't even have to count to infinity. I know he/she can, if Chuck Norris can do it, so can God. Those numbers are finite and I assume God knows everything that is finite, so if God doesn't know it then it means Fermat must be true.
Of course, Fermat is already proved, but this is just to illustrate the point.

Speaking of Chuck Norris, he can count not only to infinity, but also from infinity back to one. And in fact he has already done it. Twice.

Speaking of Chuck Norris again, we should create a Chuck Norris of figure skating. Who would that be?
Like he does octuple Axel, but he does so fast, that our eyes and cameras detect only double.
Or he is the only one who won gold medals in all four disciplines: men, ladies, pairs and dances. And he did it all by himself.
Those kind of things. It would open window for some creative humor... :)
 

Ducky

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And excuse me, the biggest art pieces were created driven by religions. :)

Because it was the Church/Monasteries/Temples/Wealthy people trying to gain favor with religious officials who were commissioning the biggest art pieces, not necessarily because the artist was inspired to create them.
 
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