Reputation
3,727
Next privilege 5,000 Rep.
Approve tag wiki edits
Badges
8 31
Newest
 Revival
Impact
~37k people reached

6h
comment If a $n$-manifold exists, then is it the boundary of an existing $(n+1)$-manifold? how can String theory state that there are only $11$ dimensions?
Your ultimate question is hard for me to figure out. If you're asking "what are the theoretical physics restrictions that lead to "11"?" then maybe physics stackexchange would be better. But when you say "the difference between theory and reality reconciled" it makes it sound like all the physics and topology is incidental to a misunderstanding you might have about formal logic or the philosophy of mathematics, but with all the details of physics and such, I can't guess the precise nature of your confusion, and would suggest trying to ask it independently of the physics, if you can.
7h
comment If a $n$-manifold exists, then is it the boundary of an existing $(n+1)$-manifold? how can String theory state that there are only $11$ dimensions?
Mathematical "existence" is very different from both physical existence and the relevant physical theoretical possibility. Without knowing the relevant physical details myself, imagine if one restriction for the string theory were something like "the number of dimensions must be squarefree" then even if the 11d candidate were the boundary of a 12d manifold, that 12d manifold couldn't be a candidate.
8h
comment If a $n$-manifold exists, then is it the boundary of an existing $(n+1)$-manifold? how can String theory state that there are only $11$ dimensions?
"How is it possible to make compatible those restrictions with [existence of 12 dimensional object]" is similar to the question "how is it possible to reconcile the complex numbers being 2 dimensional with the existence of 3d space?" The existence of some 12d spaces has no bearing on however the string theory works out.
20h
comment Are the Real numbers really Complete?
@chharvey The two properties I mentioned follow immediately from the Conway style definition (equivalence classes of pairs of sets of surrreals). And if you meant more reading about the surreals in general, there's way too much for me to put in this comment. If you're looking for a specific flavor of treatment and can't find it, maybe email me and/or post a new question.
1d
comment Can a number have an uncountably infinite amount of digits?
The examples at that mathoverflow link aren't really sets where someone does something like use more than countably many digits, because nothing like digital expansions really arise. That said, sign expansions of the Surreal numbers (or a field naturally sitting inside) are pretty close to binary expansions which can involve a an expansion of uncountable length.
Jul
29
comment Notation for polynomials and equating coefficients
Can you name the paper? Without context or an example of what the paper means by this notation, I'm not sure what "given $t$" should mean.
Jul
28
comment Permutational Question
I misread the question. Your interpretation is very likely correct. Good work.
Jul
28
comment Permutational Question
AB, AC, BA, BC, CA, CB are the six permutations of two distinct letters from {A,B,C}. Could you clarify what you're unsure of?
Jul
28
comment On a metric over m-subsets of [n]
I would say this metric is just the Manhattan Distance in $\mathbb R^m$ restricted to a very special domain.
Jul
27
comment Clarification of definition of “inverse” with quaternions
Quaternion multiplication is not matrix multiplication of the corresponding vectors, as that's undefined.
Jul
27
comment The set of all real functions of a real variable
If you can argue that $\mathfrak{c}^{\mathfrak{c}}\ge2^{\mathfrak{c}}$ then that's enough. I'm not certain where the "$2^{\aleph_0 \mathfrak{c}}$" is coming from.
Jul
20
comment Can we generalize Aleph numbers to non integer values?
I don't think there is any natural way to extend $\aleph$ to non-ordinal inputs in the surreals. Since by definition $\aleph$ makes a sort of huge jump with each successor, I have no idea what, if any, would be an appropriate way to interpolate. Certainly you could interpolate linearly or any other way you want, but I doubt this can lead anywhere satisfying.
Jul
12
comment example of multiplication of ordinals with infinite cardinality with larger value on right where we dont' take the max?
Cardinal arithmetic has addition and multiplication with an infinite cardinal reduce to max. See en.wikipedia.org/wiki/Cardinal_number#Cardinal_arithmetic (There the only case you have to watch out for is multiplication by $0$.)
Jul
11
comment Is Wolfram wrong about unique 3-colorability, or am I just confused?
Has anyone informed the website maintainers about the error?
Jul
4
revised Pseudo-Surreal numbers are analogous to?
added tags
Jul
4
answered Pseudo-Surreal numbers are analogous to?
Jul
1
comment Examples of calculus on “strange” spaces
Maybe you're looking for the term "derivation" in the sense of abstract algebra? Maybe you'd be interested in en.wikipedia.org/wiki/Arithmetic_derivative ?
Jul
1
answered Decoding the sign expansion of surreal numbers
Jul
1
comment Unusual result to the addition
@rogerl It's worth noting that that expression for a sequence of 1s can be found as an application of the formula for the sum of a finite geometric series.
Jul
1
comment Rational Irrational Numbers
As an aside, you may be interested in reading about continued fraction representations of numbers, where irrational solutions to quadratic equations with rational coefficients (like $\sqrt2$) have repeating expansions, but other irrationals don't.