/math/ - Mathematics

>>298

It feels more natural.

In ZFC, everything is a set. But what is 3 unified with pi? It makes no sense. I have never seen any argument in higher mathematics that leveraged something like this.

If we can put it all on a clean foundation, I'm all for it.

As for practicality, the higher levels of abstraction should be more or less independent of the foundations, so neither will be more practical in that sense.

>>305

Well, I don't see a problem with 3 unified with pi being a set. Pi, as a real number, is a Cauchy sequence of rational numbers, which is an countably infinite collection of ordered pairs. To me it is clear that is a set. 3 is just the collection of 0,1,2, which are all sets by the Von Neuman construction of the numbers.

3 union pi is an infinite set, and a set nonetheless. However pi is not an ordinal, (I'm not sure if it is ordinal definable or not, I think so though), so we don't have that 3 union pi is pi, as 3 union 4 would be 4.

I agree that the topos in question shouldn't affect the more abstracted results. But it seems like viewing something from a different perspective is a trend in mathematics that leads to advances.

>>306

A small note about your representation of 3 as a set: It's not incorrect, since in fact you can give it any representation you want in a number system, but it's not the same representation as the one you gave for pi.

The number 3 as an integer is actually different than 3 as a real number, but there is a natural embedding of Z into R which maps n to the sequence (n, n, n, ...), which as you say it, is the set of ordered pairs {(1, n), (2, n), (3, n), ...}.

Another small detail is that this representation is not unique, so you can fix this by defining the real number x to be the set of all Cauchy sequences converging to x. In algebra, this would be the coset denoted x + N, where N is the set of all Cauchy sequences converging to 0, with addition being element-wise addition and multiplication of the Cauchy sequences. In this sense, the union of x + N and y + N is sorta like a residue class, and although it doesn't represent a real number, I'm sure there are some pretty cool uses of it.

To answer OP's question, I'm not well-versed in category or type theory, but I think functions are a MUCH more natural first-class citizen for a more practical foundation of mathematical logic than sets. Functions have just as little structure as sets do, and they arise from nature more easily I think. This is where type theory shines, where you can disprove the existence of an object by considering the set S of all such objects and proving the existence of a function S -> {}. If S maps into the empty set, then of course S is empty, and your mathematical object doesn't exist. But whether type theory actually makes logic easier is something I'm unsure of.

>>324

Thanks for that example there at the end, really illustrates some of the power there.

However, I'm confused by your statement that functions are more fundamental first class citizens, a function requires a domain and codomain. How are these things not sets? I'm familiar with people defining constants as functions on the empty set, but even the most restricted idea of a function requires some notion of a set.

No, because the complex numbers form a field. The only ideals of a field are the ring itself and the trivial ideal, so you can't define any interesting quotient space like the integers modulo 2 or whatever.

Both, depending on how you think about it.

We have Axioms, which are things we hold to be true, but cannot prove. For instance, the idea that two parallel lines will either intersect 0 times or infinite times.

If we choose different axioms, we would still arrive to the exact same conclusions (which is a pretty good indicator that the truths are objective), but our proofs for those theorems would be radically different.

The patterns that emerge from a collection of axioms are indeed objective (in that no one else can derive different results from them) and natural (since many natural processes behave according to these patterns regardless of whether anyone has studied them.)

Choosing a set of axioms which gives a good balance of generality and usefulness is an art, so they seem more like inventions. For example, topological spaces are pretty general, so they don't have the nice properties that Hausdorff spaces have, which itself is pretty broad compared to metric spaces, which have a great deal of structure and nice properties. But Euclidean spaces, although having tons of organization, cover a very small portion of useful mathematical objects.

However, since some collections of axioms give a surprising amount of results over other less elegant attempts, one could argue that they would be invented anyway at some point, so they are discovered rather than invented.

I'm a graduate student in mathematics, and can confirm that all our perspectives, even through reason, are incapable of reaching higher reality. Sorry meat monkeys, but we're all shit stains.

As far as the "real" axioms go, it can be imagined otherwise. By that I'm talking about the axioms that are talked about today, not the commutative property of the integers. When it comes down to it, the mathematical community decides something that it decides is "cool" or is a good direction to rock with and rolls with it. Perhaps this is for lack of another viewpoint, but I don't think there are many mathematicians that will say mathematics is a fundamental property of the universe, other than from our perspective. Of course, higher level physics is based on all of this so what you have been told as true could very well be meaningless. Who cares, live your life.

>>328

At what point does a sentence arise out of mere description and into modeling?

At what point does modeling accelerate the repeatability of things?

At what point does the repeatability of things result in the influence of distant objects upon each other?

At what point must the repeatability of things be measurable through the manipulation of symbols in order for the universe to exist?

Answer those questions and you'll know if math runs the place. Nitpick, of course, is whether our current math is anything like the math the universe uses, and even so if we're constructing it correctly and completely.

>>337

A sentence doesn't model anything until it is interpreted in a model, for instance the sentence "multiplication is commutative" is meaningless until you clarify what it is you're talking about multiplying. So in some sense you have to pair the sentence with a structure in order to decide whether it has any sort of meaning beyond expressing some abstract property. I have no idea what you mean by repeatability of things.

>>320

But n(n+1)(2n+1)/6 is the formula for the sum of the first n squares.

>n(n+1)(2n+1)/6 = 200k

>n(n+1)(2n+1) = 1200k

At this stage, we know that 1200 is an upper bound for the solution.

1200 = 2^4 * 3 * 5^2

>>322

You're right, i confused the formula for the sum of the first n naturals.

The question is about distributing the prime factorization of 200 over the three elements of our final product. I am not sure how to do that systematically.

Letting n=0(mod25) and n+1=2^4*3(mod48) i find a small solution of n=575.

If we impose a restriction to the term 2n+1, I suspect that will make n grow larger. So if we can limit our restrictions to the n and n+1 terms, we can make n smaller.

112 is the answer, but I can't find a way to do this without checking all values of n with a computer. Interesting problem though.

>>322

>>323

>>325

I don't have a direct solution without a search, but the search here is over factors, not values of n

Masonic dubs...

Must post...

n(n+1)(2n+1) = 1200 does not guarantee that n is an integer

the construction of a set of solutions based on certain members, does not guarantee the solution of that set has those members

it does imply that n(n+1)(2n+1) is 1200m, a multiple of 1200

1200 is 2^4 * 3 * 5^2

1200m = 2^x*3^y*5^z*m

m can be anything that has factors spread across the three expressions

observation:

6 and 2 are even

3 and 1 are odd

even * any = even

even + odd = odd

powers of 2 are restricted to either n or n+1, at least 4 times total

powers of 3 can appear in all of them, at least once

powers of 5 can appear in all of them, at least twice

2n+1 must be odd, so for 2n+1, the power of 2 must be 0

i = n, j = n+1, k = 2n+1

so this is how far I got

n = 2^x*3^y*5^z*u, n+1 = 2^p*3^q*5*r*v, 2n+1 = 3^s*5^t*w

2^x*3^y*5^z*u = 2^p*3^q*5^r*v - 1 = (3^s*5^t*w - 1)/2

1 = 2^p*3^q*5^r*v - 2^x*3^y*5^z*u

1Post too long. Click here to view the full text.

>powers of 2 are restricted to either n or n+1, at least 4 times total

>powers of 3 can appear in all of them, at least once

>powers of 5 can appear in all of them, at least twice

I failed that

if 1200 is the target form, then 2400, 3600, 4800 are viable tests

the highest power of 2 is 4

the highest power of 3 is 1

the highest power of 5 is 2

everything else is a multiplier

back later

Op here, never mind, disregard. Fuck. Me. Patience is a virtue, and so are reading/understanding the question properly...

Top of /prog

>>56

I think there's a pretty big misconception that your post embodies.

>I remember when I got into computers

you weren't doing computer science, you were building a computer.

Math is similar. Math doesn't actually tell you how to build a bridge or a rocket ship, that's physics and engineering, which apply mathematical concepts but aren't actually math. Nobody does actual math until they get to calculus, and by that time they've been trained by the bullshit of high school math to only look for the answer, instead of seeing the innate beauty of the process. Only a few do.

Because high school - undergrad math is tedious and boring unless you have the autism to remember and practice everything you'll need for higher level math (Number theory, combinatorics etc.)

>>256

Dude that's general ignorance. I fucking blow at math but a LOT of people don't know much aside from basic arithmetic. It's just not used in their lives.

It's like English, they use what they have as best they can. They're not out to write books.

https://www.maa.org/external_archive/devlin/LockhartsLament.pdf

Factoring

Equations

Functions

Graphs of Functions

And really at this stage, they should know some trig, even though it's not essential to finding quadratic roots.

x = 1 + x/2

2x = 2 + x

2x - x = 2

x = 2

>>251

Is it the golden ratio or is it just 1.5?

>>251

>the value of this bill is 1

okay x=1

>plus half its value

if his value is 1, and half 1 is 0.5

1+0.5=1.5

>>314

Contradiction.

x cannot be both 1 and 1.5.

>>309

Neither - it's 2.