/sci/ - Science & Math

whats the difference?

Math is true by definition. We define it, so that's how it is.

I think nature is just being the awesome motherfucker that it is.

Math is not essential. Neither is logic. They are simply constructs that allow us to perceive the world.

I honestly believe we've been able to describe it pretty accurately with mathematics, and it is a language of description to help us intellectually grasp something using logic. But whenever you go into the quantum world, math gets way beyond fucked up and we can hardly describe what happening

>>2601390
In the former the universe is based on mathematics. In the latter math is abstract and separate from the universe.

>>2601431
Very philosophical. What's the observable difference?

>>2601450
There are some phenomena that might be impossible to accurately describe mathematically.

>>2601431
If math were abstract and separate from the universe, it wouldn't make sense that the universe so closely follows mathematical principles.

>>2601420
Am I the only one that saw this?
Someone put a legit name and e-mail on 4chan? Are you this guy?
http://mollie.berkeley.edu/~taylor/people/ce.html

>>2601532
thats a pretty generic name dumbass, there are probably hundreds of people with that name.

In a sense, math would exist without the universe, provided deductive logic functioned, and axioms like a=a hold true. But, those basic principles might be related to the nature of the universe, as evidenced by the way Nature tends to not only be describable in mathematical terms, but be describable in amazingly simple ways. Also, there are very, very few pieces of even the most abstract mathematics that have no place in describing the universe. I believe the two are very fundamentally related.

>>2601379
>have we just managed to use mathematics to very accurately describe physical phenomena?
I subscribe to this philosophy, and/or to a philosophy from which that follows.

>>2601387
Is awesome.

>>2601387

No, math is synthetic a priori.

If it were analytic, then "A triangle has three angles that sum to 180 degrees" would be true by definition alone. But, it can be false that the angles of a triangle sum to 180, e.g. in a non euclidean space.

you guys are fags.

>>2601759
I hope you're really not that stupid.

>>2601759

True, but non euclidean spaces could be in the "set" of things conceivable by the human mind. A human mind created by this universe.

I'm of the belief that mathematics (language) is discovered, not invented.

>>2601792

I'm approx 98.4% smarter than you.

>>2601811

I really can't tell what your argument is, since what it looks like is blatantly non-sequitur.

Everyone interested in this thread needs to learn about fractals

>>2601759

I can prove you wrong.

Algorithms either halt or run forever, this is a given. A "hello world" program will definitely halt (if it's properly written).

10 GOTO 20
20 GOTO 10

Will run as long as your computer is working. Guaranteed.

We can write an algorithm at random; every string of binary data represents a valid machine-coded program in some system.

Most of these programs will do essentially nothing, but they will all either halt at some point in the future or never halt. There's no in between.

Therefore, there's some probability that an algorithm I generate at random will halt. That probability is a number, determined by the algorithm coding. It definitely exists; it's a valid description of the physical system which is running the algorithm.

However, that halting probability cannot be calculated. It can only be discovered, it can never be generated synthetically.

Therefore u==incorrect.

Mathematics is simply the language of exactitude. A little while ago, the human race figured out that simply calling things "large" or "small," or describing something as "in many" or "in few" amounts was completely pointless after a certain point, and measurability and accountability required something more quantitative. You can surely say math is "true," insofar as a ruler gives a "true" measurement of a block of wood. The block of wood is certainly whatever size it is, but centimeters and inches are simply our expressions of the size. Math works in this way, it's our expression of describing complicated mechanics that could not be explained efficiently using words .Try to imagine fully grasping a physics course without math involved. It'd be as useful as psychology is today, and feats such as space exploration would never become possible. So, they are "true," but as a note; The effectiveness of mathematics gets drawn down the more complicated the math becomes, as we inevitably encounter more and more missing and virtually unaccountable variables. But those limitations and the strive for higher mathematical knowledge is one in the same. Happy exploring gents.

>>2602007
<3 Theory Of Computation.
And thus I really <3 you.
Thank you.

I'm not really sure that refutes anything he said though, but bravo anyway.

>>2602007

It's a lot easier for me to prove you wrong, on account of the fact that you don't know what 'synthetic' means.

Analytic statements = statements in which the relation between the subject and predicate is identity or partial identity, and therefore the truth of which can be ascertained simply in virtue of knowing the definition of terms. (e.g. All bachelors are married men)

Synthetic statements = statements the truth of which cannot be ascertained in virtue of the meaning of terms alone (e.g. "...that halting probability cannot be calculated...")

woah dude. Of course the universe follows "mathematical rules" mathematics is logic, which is the law of existence itself. If anything exists it follows mathematical rules.

>>2602031
The way I use the term, mathematics and the languages used to describe mathematics are two different things.

>>2601379
I'd say you should watch "The Life of Chaos." Its a pretty interesting documentary and related to the subject somewhat.

>>2602064
Correction: I think it was named "The Secret Life of Chaos" actually. My bad

>>2601431
Math is clearly abstract to me. You can have any math you want, but most of these systems would be completely useless to describe our universe.

>>2602051
>>2602051

I'm sorry, what I meant to say is that it cannot be synthetic A PRIORI. Rather, mathematics is synthetic a posteriori.

Glad you pointed that out though, I could have looked like a total fool.

>>2601960
>>2601759
>>2602051

It's pretty clear that you're good at repeating things your 20th Century Philosophy professor told you.

Stop pretending to understand maths philosophy fag.

>Does the universe follow mathematical rules, or have we just managed to use mathematics to very accurately describe physical phenomena?

The latter.

Math is a tool of logic we created. The universe does not give a flying fuck about it, or anything else we think for that matter.

>>2602146

Here here. Restating Kant 101 shit is pretty weak.

>>2602148
>The universe does not give a flying fuck about it, or anything else we think for that matter.
WE'LL HAVE TO FUCKING MAKE IT CARE THEN WONT WE.

>>2602162

I look forward to that day.

>>2602148

As I showed above... >>2602007

Mathematics is not just a descriptive tool. It describes objects which must be empirically observed. The behaviour of algorithms can be observed but not generated by a prior assumptions, therefore the structures described by mathematics have an empirical existence outside of matter.

>>2602109
>>2602146

What would Kripke say? Wouldn't he say that mathematical concepts are essential and that though investigation their attributes are revealed?

I think I am against this. I tend to think that mathematics is a construct. The nature of which is due to human neural architecture and cognitive abilities. It is simply a tool that we have invented to help us live in the world we are able to perceive/conceive.

>>2602166
>mfw we can move entire galaxies at whim

>>2602170
Ok, let me try and parse this.

>Mathematics is not just a descriptive tool.
I think I disagree.

>It <span class="math"> describes objects which must be empirically observed.
No. My high school calc teacher invented his own geometry. It was consistent, non-Euclidean, non-trivial, and as far as he knew entirely useless to describe the natural world.

Also, if there are less than 2^100 fundamental particles in the observable universe, does that mean that I cannot talk about "2^100 + 2^100"? But I can. Thus I can talk about math independent of observations in the natural world.

>The behaviour of algorithms can be observed but not generated by a prior assumptions,
Not sure what you're getting at here. A Turing machine is a formalization of an algorithm, formalizing using basic logic. One can use math to make proofs about Turing machines, but the Halting problem is not solvable.

>therefore the structures described by mathematics have an empirical existence outside of matter.
As I don't understand what you said before, I don't understand this.

Also, what does "empirical" mean in this context?

http://en.wikipedia.org/wiki/Empirical
>The word empirical denotes information gained by means of observation, experience, or experiment.

I think you're being inconsistent here. You're saying it's both not-observable, but observable.[/spoiler]

>>2602170

congratulations you proved math is essential by using a machine that was built to follow math to follow math

>>2602170
Ok, let me try and parse this.

>Mathematics is not just a descriptive tool.
I think I disagree.

>It (math) describes objects which must be empirically observed.
No. My high school calc teacher invented his own geometry. It was consistent, non-Euclidean, non-trivial, and as far as he knew entirely useless to describe the natural world.

Also, if there are less than 2^100 fundamental particles in the observable universe, does that mean that I cannot talk about "2^100 + 2^100"? But I can. Thus I can talk about math independent of observations in the natural world.

>The behaviour of algorithms can be observed but not generated by a prior assumptions,
Not sure what you're getting at here. A Turing machine is a formalization of an algorithm, formalizing using basic logic. One can use math to make proofs about Turing machines, but the Halting problem is not solvable.

>therefore the structures described by mathematics have an empirical existence outside of matter.
As I don't understand what you said before, I don't understand this.

Also, what does "empirical" mean in this context?

http://en.wikipedia.org/wiki/Empirical
>The word empirical denotes information gained by means of observation, experience, or experiment.

I think you're being inconsistent here. You're saying it's both not-observable, but observable.

>>2602180

You appeal to a "construct" of "human neural architecture".

The word "construct" is an inherently mathematical word. So is architecture.

Face it. The ONLY way that one can interface with the universe is mathematically. What we see and feel and think are patterns of energy in our nervous systems.

Patterns require some substrate, and that substrate is what we refer to as mathematics. Patterns are inherently mathematical. The fact that there are predictable events in our universe proves that physical phenomena are a manifestation of mathematics, and not the other way around.

>>2602170
>Mathematics is not just a descriptive tool
>It describes objects
>derp

Are you mental? The universe do not follow our math. Our math is modeled after it.

>>2602109

I was being faceitous -- hardly anyone really agrees with Kant. But your argument still fails to show that math isn't a priori. Those people who think that math is don't deny that math can be known a posteriori. Instead they think that mathematical truth is a matter of necessity as opposed to being contingent on facts about the empirical world. But if mathematical truth isn't fixed by facts about the empirical world, then it is possible to know mathematical truth without knowing of the empirical world.

Presumably the point at which your program terminates follows necessarily from the initial set-up and the syntactical rules of the system. If not, then there's some genuine indetermination in the system. But that would mean it would simply follow statistical mathematical models, which aren't inconsistent with a priori views of math either.

>>2602146

Stop pretending you understand the philosophy of math, you math fag. As if they teach an answer to this question in any math course.

>>2602208

as a late 20th century capitalist human from the cultural west of course I use mathematical concepts like construct and architecture. Does this mean that outside of my own though styles and those of people like me that math as we know it is essential. Does it mean that beyond human perception that math is essential. Does of concept of mathematics correspond on a 1:1 basis with "essential" math. I think that I am rightly skeptical about it.

>>2602203

>Ok, let me try and parse this.

No need to get snippy. I was perfectly cogent.

>It (math) describes objects which must be empirically observed.
>No. My high school calc teacher invented his own geometry. It was consistent, non-Euclidean, non-trivial, and as far as he knew entirely useless to describe the natural world.
See, the key here is that It describes things which must be observed; not that EVERYTHING it describes can be observed.

"There exist some mathematical structures which must be observed physically to be understood" is not equivalent to "All mathematical structures can be observed physically"

>Not sure what you're getting at here. A Turing machine is a formalization of an algorithm, formalizing using basic logic. One can use math to make proofs about Turing machines, but the Halting problem is not solvable.
I'll agree, but I wasn't talking about Turing machines. They describe algorithms, they are not algorithms themselves. Algorithms have a separate existence.

>As I don't understand what you said before, I don't understand this.

That's certainly not my problem. I was very clear.

>Also, what does "empirical" mean in this context?
>I think you're being inconsistent here. You're saying it's both not-observable, but observable.
I said nothing of the sort. What I said was that it's not calculable, but it is observable.

In the sciences, it is understood that one can never measure something with perfect accuracy. In mathematics, typically EVERYTHING is described with perfect accuracy.

The halting probability can be measured; we can run programs for ever and ever and measure it to finer and finer details. In fact, in order to determine it it has to be measured; it can not be derived except by actually performing the algorithms themselves BECAUSE of the undecidability of the halting problem.

Therefore it must be OBSERVED, therefore it is empirical.

>>2602227

Just because you use base 10 and a computer uses base 2 does not mean you are describing different things. It's simply a matter of language; the underlying structure they describe is the same.

All valid mathematical models agree. Our mathematical model is valid, as it describes the universe very well. Therefore any other civilization, on the earth or elsewhere, which has developed a different mathematical system, will agree with ours. It's merely a matter of translating between the systems.

>>2602253
>The halting probability can be measured; we can run programs for ever and ever and measure it to finer and finer details. In fact, in order to determine it it has to be measured; it can not be derived except by actually performing the algorithms themselves BECAUSE of the undecidability of the halting problem.

>Therefore it must be OBSERVED, therefore it is empirical.

You must not have been paying attention in Theory Of Computation that day. The proportion of Turing machines which halt is not computable. There is no finite process or algorithm which can find it. Your little trick of "measuring it in the real world" doesn't get around this particular problem this time either. It's not computable, and it's not measurable.

>>2602262

>is having trouble with the concept of inscrutability

>>2602253
>I'll agree, but I wasn't talking about Turing machines. They describe algorithms, they are not algorithms themselves. Algorithms have a separate existence.

The only working definitions of existence which I have are:
1- The existence of observable non-abstract, things.
2- The existence of (abstract) solutions to (abstract) math constraints.

What does it mean for an algorithm to exist?

>>2602210

In the sense that it can not be obtained by formal mathematics, it is not a priori. No formal juggling can produce the halting probability.

Let me say that again. It is proveably impossible to determine the halting probability in a mathematical system with finite axioms.

This means that the halting probability is not just a synthetic result of a priori assumptions (read: axioms), but rather has to be determined by a posteriori means (read: physically running the algorithms).

This follows automatically from Godel's Incompleteness theorem, and especially from the work of Gregory Chaitin.

>>2602271
>no finitely describable process
fixed

>>2602282
>(read: physically running the algorithms).
But you can't even get it by running the algorithms. There is no finitely describable process to determine if an algorithm halts or not. Thus your idea breaks down. You can't just say "run it" if you have no ability to recognize a halting machine from a non-halting machine through observation.

>>2602271

Measured == Measured within some observational error.

Just as you cannot have a perfect ruler, you can't have a perfect measurement of the halting probability. But you can measure it to within some error by brute force; the error goes down as you check more algorithms.

>>2602295
But again, you cannot even measure it because you lack a way to distinguish a halting machine from a non-halting machine through observation in a finite amount of time.

>>2602278

Patterns exist. They are observable as much as an atom (which is also a pattern of energy), or a rock (which is a pattern of matter (which is a pattern of energy)) or you (ditto).

Patterns are inherently mathematical. They cannot exist without mathematical rules to form them. And if they exist empirically, the rules must also.

>>2602300
What is a pattern? A model, or description, in a formalized logic that can give falsifiable predictions? Are you trying to claim something more than that?

>>2602290
>>2602284
>>2602271

Read my response here: >>2602295

Apparently none of you have any idea what measurement means.

I UNDERSTAND THE FACT THAT THE HALTING PROBABILITY IS NOT COMPUTABLE. THAT IS ACTUALLY THE CRUX OF MY ARGUMENT.

Let's say I run a hundred algorithms. After 100 years, 60% of them halt. I have made a measurement with an observational error which is also not computable.

I have, however, measured the lower bounds of the halting probability with practically no error. That is a measurement of an empirical process. I can't get that formally out of the axioms.

>>2602304
Let me simplify the question. The number 5. To me, the number 5 is a human convention, an abstraction, in the formalized human logic system of ZFC. I can use the number 5 as part of a scientific theory to model the physical world, and provide falsifiable predictions.

Are you arguing that 5 exists in some other sense?

>>2602305
>I run 100 arbitrary programs.
>After 100 years, 60% halt.
>Lower bound.
Uh dude, are you really that retarded? I'll give you a chance to correct your error.

>>2602297
You can recognize those that *are* halting machines and those that don't halt below a certain number of steps, though. As you increase the number of steps you check, you gain more and more accuracy.
It's no different from the fact that there are infinitely many programs to check; you just have to fight against two infinities for your estimation.

>>2602307

Numbers are patterns. If the number 5 did not have a physical existence, the universe would not distinguish between 5 Joules and 4 Joules and there could be no conservation of energy.

However, there is an observable, empirical conservation of energy, and so numbers must have some physical existence.

I would also argue that an atom exists, even though it's just a pattern of energy. And energy exists even though it's a pattern of motion or matter or whatever else. EVERYTHING is patterns.

>>2602224

Actually, I'm a physics fag. And I doubt you've taken any graduate math courses or keep up with the literature.

Listen guys, I'm tired of hearing this non-euclidean geometry argument - it's bogus. This is NEW MATHEMATICS (In the grand scheme of things), just because we haven't found a physical system which can be modeled with a non-euclidean geometry doesn't mean one doesn't exist.

For example, topology is a little bit older than non-euclidean space and we're just starting to wonder if it can be used to describe quantum entanglement.

Math is discovered and usually 150+ years later we figure out that its pretty damn close to some physical system which leads me to believe that math is determined by the universe you exist in.

>>2602311
And the answer:

This is your argument:
>Let's say I pick the first 100 integers. I test them all for prime-ness. I find that X are prime. Thus the lower bound on the fraction of integers which are prime is X / 100.

See a problem here?

Your idea might work if you could run all Turing machines at once, but of course that requires infinite mass, space, and (setup) time.

It's not computable. It's provably not computable. It's not even approximatable. Give it up, and go back to school. Us CS experts need to have some grown up talks.

>>2602311

Very kind of you.

60% would be the lower bound for THAT set.

The lower bound of the halting probability can be determined for sets of programs (i.e. under a certain number of tokens), and as you increase the tokens you decrease your error.

As somebody else already argued.

>>2602325
>and as you increase the tokens you decrease your error.
By tokens, you mean number of internal states? Also, proof please.

>>2602323

>Your idea might work if you could run all Turing machines at once, but of course that requires infinite mass, space, and (setup) time.

>It's not computable. It's provably not computable. It's not even approximatable. Give it up, and go back to school. Us CS experts need to have some grown up talks.

You CS fags clearly do not understand how real science is conducted. You can determine boundaries for a measurement in science, but the measurement with arbitrary error is impossible to determine in finite time.

What I determine in this experiment is the lower boundary of the halting probability for that set of programs. This number, as is the halting probability of all programs, is not calculable. But I can measure it empirically.

Read up on Chaitin's constant and get back to me.

>>2602325
Specifically, I remain unconvinced that your scheme will work. Do you have a proof for it? What arguments have you that this approximation scheme converges on the actual halting fraction?

>>2602323

Implying this is applicable argument for any field other than analytical philosophy

>>2602328

I am not currently talking about Turing machines, but rather about computer programs.

By tokens I mean a single, imperative statement. But you can use states in a turing machine if you like, it'll work out the same.

And as for proof:

If 60 of the programs have halted after a hundred years, that is the MINIMUM number of programs which halt out of that set. If I run those programs again, they won't suddenly stop halting; they will always halt.

However, it's possible that some of the other 40 programs that have not halted will someday halt. So if 2 more programs have halted after 200 years, my new lower bound is 62%. This has to be higher than the original number, it'll never get lower, and so it is in fact a lower bound.

Therefore the lower bound will approach the actual probability as time approaches infinity.

>>2602338
I'd like to point out the flaw in your argument. More like paradox.

If your scheme works, then it is a finitely describable algorithm which can compute the number to an arbitrary degree of accuracy. Thus Chaitin's constant is computable. However, it has been proved that it's not computable, which means that there must be some flaw in your scheme.

The scheme will only converge on the halting probability for that set of programs. However, that halting probability is also not calculable.

I proved it will converge here: >>2602359

Too tired to bother with a more formal proof, but it should be immediately obvious to anybody with any mathematical training.

>>2602360
See:
http://en.wikipedia.org/wiki/Chaitin%27s_constant
>Super Omega
>As mentioned above, the first n bits of Gregory Chaitin's constant Omega are random or incompressible in the sense that we cannot compute them by a halting algorithm with fewer than n-O(1) bits. However, consider the short but never halting algorithm which systematically lists and runs all possible programs; whenever one of them halts its probability gets added to the output (initialized by zero). After finite time the first n bits of the output will never change any more (it does not matter that this time itself is not computable by a halting program). So there is a short non-halting algorithm whose output converges (after finite time) onto the first n bits of Omega. In other words, the enumerable first n bits of Omega are highly compressible in the sense that they are limit-computable by a very short algorithm; they are not random with respect to the set of enumerating algorithms. Jürgen Schmidhuber (2000) constructed a limit-computable "Super Omega" which in a sense is much more random than the original limit-computable Omega, as one cannot significantly compress the Super Omega by any enumerating non-halting algorithm.

>>2601379
I think the universe is an instatiation of some mathematical structure and if it wasn't, we probably wouldn't exist at all.

However, a mathematical structure does not mean that it's identical to our math, it merely means that our math can be used to build a model which describes it perfectly and is isomorphic to whatever the structure really is.

Some other interesting things on this subject:
http://arxiv.org/abs/arXiv:gr-qc/9704009

>>2602360

I've mispoken a couple of times, the consequences of being up all night and having to respond so rapidly. I apologize for that.

The number is not computable, and I still agree with that.

Let's look at it simply. The halting problem is decidable for the set of programs which contain no conditional loops, which should be obvious. This sets a lower boundary on the halting probability.

I can determine the percentage of programs which will never contain any looping tokens, and thus could not have loops. let's say this is 15%. That's a lower boundary on the halting probability; I can prove that without loops a program must halt, so therefore that 15% will never halt.

Therefore the halting probability is AT LEAST 15%.

To determine the probability with complete accuracy would require infinite axioms.

>>2602282

Hi and Godel believed mathematical truth was a priori.

>>2602373

Thank you sir. Or madam.

>>2602404
Just like the Busy Beaver sequence is not computable - which means that there is no finitely describable program which can calculate the nth term, but some of the initial terms have been computed. (My prof back in college was part of the group which calculated the 6th busy beaver number IIRC.)

Similarly, Omega is not computable, which means you cannot give a finitely describable algorithm which computes it to an arbitrary degree of precision, but you can compute some of its initial bits.

>>2602406

I have no idea where you got that idea from.

Godel was a platonist. He believed there was a fundamental mathematical reality which was discovered by the mind. This is a posteriori.

>>2602426

Platonists do not believe mathematical truth is a posteriori. Platonic truths are necessary and can be known without examining the world.

>>2602445

It depends on how we define the world. The mathematical world can be known without knowledge of the physical world. In that sense it is a priori.

But Platonic realism states that mathematical truths have a real existence outside of physical reality. They encompass another world. In the case of mathematical structures which must be observed and cannot be generated formally, this is a posteriori knowledge gained from direct contact with the mathematical world.

Thus it depends on your frame of reference.

Now that this argument has died down, and nobody has successfully refuted my position that mathematics is fundamentally real,

I declare myself the winner. Goodnight everybody.

>>2602496
Sure, for whatever unformalized definition of real.

>>2602517

Formalizations which can describe reality are necessarily incomplete or inconsistent, because they are capable of describing arithmetic (as per Godel's Incompleteness Theorems)

My real is an undefinable, but nonetheless empirical, real.

>>2602496
As long as by real you don't mean that real numbers are involved as I have some severe doubts that non-discrete worlds are possible. I have a much higher degree of confidence in a discrete world's existence than in a world which can only be described in reals.

does scientist have this argument every night?

>>2602588

He does. It quite amusing to have seen him start as a hard quine and grow more and more grokish with each argument.

>>2601759
That shows it's not a priori, bud.

>>2603328
I see you corrected that. Sorry.

I don't understand why these arguments don't instantly turn to geometry rather than arithmetic.

the map is not the territory

>>2603343
How can you know? While QM and GR may not be complete enough to describe reality, when we do find a theory isomorphic to reality, it'll be just as good, and surely since the world works, there is some mathematical structure underneath it all, we'll just never know exactly what it is, the best we can do is find something isomorphic to it and use that to make predictions.