/sci/ - Science & Math

There are things that are discovered, and things that are invented. The boundary is put at different places by different people. I put myself on the list and I believe that my position is objectively justifiable, and others are not.
>Definitely discovered: finite stuff
By probablistic considerations, I am sure that nobody in the history of the Earth has ever done the following multiplication: 9306781264114085423 x 39204667242145673 = ?
Then if I compute it, am I inventing it's value, or discovering the value? The meaning of the word "invent" and "discover" are a little unclear, but usually one says discover when there are certain properties: does the value have independent unique qualities that we know ahead of time (like being odd)? Is it possible to get two different answers and consider both correct? etc.
In this case, everyone would agree the value is discovered, since we actually can do the computation--- and not a single (sane) person thinks that the answer is made up nonsense, or that it wouldn't be the number of boxes in the rectangle with appropriate sides, etc.
There are many unsolved problems in this finite category, so it isn't trivial:
>Is chess won for white, won for black, or a draw, in perfect play?
>What are the longest possible Piraha sentences with no proper names?
>What is the length of the shortest proof in ZF of the Prime Number Theorem? Approximately?
>What is the list of 50 crossing knots?
You can go on forever, as most interesting mathematical problems are interesting in the finite domain too.

>Discovered: asymptotic computation
Consider now an arbitrary computer program, and whether it halts or does not halt. This is the problem of what are called "Pi-0-1 arithmetic sentences" in first order logic, but I prefer the entirely equivalent formulation in terms of halting computer programs, as logic jargon is less accessible than programming jargon.
Given a definite computer program P written in C (or some other Turing complete language) suitably modified to allow arbitrarily large memory. Does this program return an answer in finite time, or run forever? This includes a hefty chunk of the most famous mathematical conjectures, I list a few:
>The Riemann hypothesis (in suitable formulation)
>The Goldbach conjecture.
>The Odd perfect number conjecture
>Diophantine equations (like Fermat's last theorem)
>consistency of ZF (or any other first order set of axioms)
>Knesser-Poulson conjecture on sphere-rearrangement
You can believe one of the two
>Does P halt" is absolutely meaningful, so that one can know that it is true or false without knowing which.
>"Does P halt" only becomes meaningful upon the halting of P, or a proof that it doesn't halt in a suitable formal system, so that it is useful to introduce a category of "unknown" for this question, and the "unknown" category might not eventually become empty, as it does in the finite problem case.
Here is where the intuitionists stop. The famous name here is
>L.E.J. Brouwer
The intuitionistic logic is developed to deal with cases where there are questions whose answer is not determined true or false, so that one cannot decide the law of excluded middle. This position leaves open the possibility that some computer programs that don't halt are just too hard to prove halt, and there is no mechanism for doing so.
While intuitionism is useful for situations of imperfect knowledge (like us, always), this is not the place where most mathematicians stop.

>Most believe discovered: Arithmetic heirarchy
There are questions in mathematics which cannot be phrases as the non-halting of a computer program, at least not without modification of the concept of "program". These include
>The twin prime conjecture
>The transcedence of e+pi.
To check these questions, you need to run through cases, where at each point you have to check where a computer program halts. This means you need to know infinitely many programs halt. For example, to know there are infinitely many twin primes, you need to show that the program that looks for twin primes starting at each found pair will halt on the next found pair. For the transcendence question, you have to run through all polynomials, calculate the roots, and show that eventually they are different from e+pi.
These questions are at the next level of the arithmetic heirarchy. Their computational formulation is again more intuitive--- they correspond to the halting problem for a computer which has access to the solution of the ordinary halting problem.
You can go up the arithmetic hierarchy, and the sentences which express the conjectures on the arithmetic hierarchy at any finite level are those of Peano Arithmetic.
There are those who believe that Peano Arithemtic is the proper foundations, and these arithemtically minded people will stop at the end of the arithemtic hierarchy. I suppose one could place Kronecker here:
>Leopold Kronecker: "God created the natural numbers, all else is the work of man."

To assume that the sentences on the arithmetic hierarchy are absolute, but no others, is a possible position. If you include axioms of induction on these statements, you get the theory of Peano Arithmetic, which has an ordinal complexity which is completely understood since Gentzen, and it is described by the ordinal epsilon-naught. Epsilon-naught is very concrete, but I have seen recent arguments that it might not be well founded! This is completely ridiculous to anyone who knows epsilon-naught, and the idea might strike future generations as equally silly as the idea that the number of sand grains in a sphere the size of Earth's orbit is infinite--- an idea explicitly refuted in "The Sand Reckoner" by Archimedes.

>Most believe discovered: Hyperarithmetic heirarchy
The hyperarithmetic hierarchy is often phrased in terms of second order arithmetic, but I prefer to state it computationally.
Suppose I give you all the solution to the halting problem at all the levels of the arithmetic hierarchy, and you concatenate them into one infinite CD-ROM which contains the solution to all of these simultaneously. Than the halting problem with this CD-ROM (the complete arithmetic-hierarchy halting oracle) defines a new halting problem--- the omega-th jump of 0 in recursion theory jargon, or just the omega-oracle.
You can iterate the oracles up the ordinal list, and produce ever more complex halting problems. You might believe this is meaningful for any ordinals which produce a tape.
There are various stopping points along the hyperarithmetic hierarchy, which are usually labelled by their second-order arithemtic version (which I don't know how to translate). These positions are not natural stopping points for anybody.

>Church Kleene ordinal
I am here. Everything less than this, I accept, everything beyond this, I consider objectively invented. The reason is that the Church-Kleene ordinal is the limit of all countable computable ordinals. This is the position of the computational foundations, and it was essentially the position of the Soviet school. People I would put here include
>Yuri Manin
>Paul Cohen
In the case of Paul Cohen, I am not sure. The ordinals below Church Kleene are all those that we can definitely represent on a computer, and work with, and any higher conception is suspect.

>First uncountable ordinal
If you make an axiomatic set theory with power set, you can define the union of all countable ordinals, and this is the first uncountable ordinal. Some people stop here, rejecting uncountable sets, like the set of real numbers, as inventions.
This is a very similar position to mine, held by people at the turn of the 20th century, who accepted countable infinity, but not uncountable infinity. Those who were here include many famous mathematicians
>Thorvald Skolem
Skolem's theorem was an attempt to convince mathematicians that mathematics was countable.
I should point out that the Church Kleene ordinal was not defined until the 1940s, so this was the closest position to the computational one available in the early half of the 20th century.

>Continuum
Most practically minded mathematicians stop here. They become wary of constructions like the set of all functions on the real line, since these spaces are too large for intuition to comfortably handle. There is no formal foundation school that stops at the continuum, it is just a place where people stop being comfortable in absoluteness of mathematical truth.
The continuum has questions which are known to be undecidable by methods which are persuasive that it is a vagueness in the set concept at this point, not in the axiom system.

>First Inaccessible Cardinal
This place is where most Platonists stop. Everything below this is described by ZFC. I think the most famous person here is:
>Saharon Shelah
I assume this is his platonic universe, since he say so explicitly in an intro to one of his more famous early papers. He might have changed his mind since.

>Infinitely many Woodin Cardinals
This is the place where people who like projective determinacy stop.
It is likely that determinacy advocates believe in the consistency of determinacy, and this gives them evidence for consistency of Woodin Cardinals (although their argument is somewhat theological sounding without the proper computational justification in terms of an impossibly sophisticated countable computable ordinal which serves as the proof theory for this)
This includes
>Hugh Woodin

>Possibly invented: Rank-into-Rank axioms
I copied this from the Wikipedia page, these are the largest large cardinals mathematicians have considered to date. This is probably where most logicians stop, but they are wary of possible contradiction.
These axioms are reflection axioms, they make the set-theoretic model self-simialar in complicated ways at large places. The structure of the models is enormously rich, and I have no intuition at all, as I barely know the definition (I just read it on Wiki).

>Invented: Reinhard Cardinal
This is the limit of nearly all practicing mathematicians, since these have been shown to be inconsistent, at least using the axiom of choice. Since most of the structure of set theory is made very elegant with choice, and the anti-choice arguments are not usually related to the Godel-style large-cardinal assumptions, people assume Reinhardt Cardinals are inconsistent.
I assume that nearly all working mathematicians consider Reinhardt Cardinals as imaginary entities, that they are invention, and an inconsistent invention at that.

>Definitely invented: Set of all sets
This level is the highest of all, in the traditional ordering, and this is where people started at the end of the 19th century. The intuitive set
>The set of all sets
>The ordinal limit of all ordinals
These ideas were shown to be inconsistent by Cantor, using a simple argument (consider the ordinal limit plus one, or the power set of the set of all sets). The paradoxes were popularized and sharpened by Russell, then resolved by Whitehead and Russell, Hilbert, Godel, and Zermelo, using axiomatic approaches that denied this object.
Everyone agrees this stuff is invented.

>>11308725
>>11308715
Mathematical methods and operations are invented.
Mathematical statements, relationships, and truths are all discovered.

Addition is a man-made construct, but 2+2=4 is a mathematical truth that we discovered using addition as a tool.

>>11308892
>Mathematical methods and operations are invented.
>Mathematical statements, relationships, and truths are all discovered.
incorrect, see the posts above yours

>>11308725
>>11308735
>>11308751
>>11308754
>>11308761
>>11308762
>>11308767
>>11308770
>>11308773
>>11308779
>>11308784
>>11308788
>>11308789
very reasonable writeup, if you want to know my opinion:
arithmetic hierarchy: discovered
first uncountable ordinal and continuum: weird and gets even weirder the more you think and read about it (as in measure theory stuff, axiom of determinacy), but it's useful so no one really opposes it
Using these as a tool feels okay, but studying measure theory for its own sake seems dubious
first inaccessible cardinal: we should definitely stop here, or maybe somewhere earlier, and I believe most non-set-theoretic mathematicians would agree, if they knew what is a strongly inaccessible cardinal.
everything from now here onwards: I barely have a clue, but I view it as a study of "what do different set theoretic axioms imply?" rather than a study of some actual objects

>>11308725
Classical math (the foundations): discovered
New hipster math like the category theory: invented

>grug see apple
>grug see another apple
>grug has dual apples
>grug gives whole apple to grugerina
>grug has whole apple

>>11309428
Why do you shit on Category Theory in every thread?

>>11309474
cause it is just a bunch of arrows for hipsters
not in every thread, just in math threads

>>11308715
Useless math is invented

>>11308725
Ron Maimon copy pasta?

>>11308715
Mathematics was invented, it's implications discovered.

bump

>>11308715
math is just a description of order, which is actual

Question is - do we get to recognize order as it exists in the world, or do we just reflect on our own information-organizing faculties and mistake that for external discovery?

Only the philosopher can tell you these things

>>11308715
Same thing

>>11308715
high iq men discover math, low iq men invent math.

>>11312435
no not the same thing retard. the question is: would intelligent beings evolving 10 billion light years away from us come to the same conclusion?
if so math really is unlocking this special part of the universe that gives rise to intelligence in it.
if not, it's just bullshit and mental masturbation and people who follow the "invented" kind of math (math that's clearly contradictory in the real world) are usually not just low iq but also very malignant on pushing the worthless creation of their minds on other people.
only a low iq scumbag would say it's the same thing or it doesn't matter. it fucking matters.
calculating square root 2 is something every intelligent civilization in the universe will need at some point.
mental masturbation is something else.

>>11312474
underrated post

>>11308715
Who gives a shit? It’s just semantics.

Math is discovered.
Notation is invented

>>11312474
nice! also this!

>>11308751
>they correspond to the halting problem for a computer which has access to the solution of the ordinary halting problem.

how can a computer that has access to the solution of the "ordinary" halting problem, not also have access to the solution of these problems?

worst case, keep generating code for a turing machine (on a universal turing machine interpreter), that finds if there are larger twin primes than n, looping through all positive integers.

something like this for ram machines in pseudcode:

i = 0;
while(true)
if (DoesItHalt(GenerateProgramCodeForFunctionThatFindsIfThereAreLargerTwinPrimesThanN(i)))
halt;
else
i = i + 1;

>>11313044
then run DoesItHalt on that program code, and have the answer.

>>11313044

HaltsIfThereAreFiniteNumberOfTwinPrimes:

i = 0;
while(true)
if (!DoesItHalt(GenerateProgramCodeForFunctionThatFindsIfThereAreLargerTwinPrimesThanN(i)))
halt;
else
i = i + 1;


then running DoesItHalt(HaltsIfThereAreFiniteNumberOfTwinPrimes)

(code corrected :^))

what is a halting problem?

>>11308761
>Than the halting problem with this CD-ROM (the complete arithmetic-hierarchy halting oracle) defines a new halting problem
can you give an example

bump

>>11308715
Did you make this image? I like.

>>11316289
yes it is mine

>>11316651
I will honor you by swiping it for future use if you do not object. And art imitates life, I guess math is art.

>>11316767
d'nont