@All, I'm just curious if it's normal to want a poster of 31399719737866347113914486515772694858917594191229 38744591877656925789747974914319422889611373939731 on your wall? I heard puberty was, by far, a very bizarre experience... But, I have not recalled the want of particular numbers posted on your wall as part of said experience.
What about the reversible 100 digit prime (i.e. it is a prime if written backwards)
$31399719737866347113914486515772694858917594191229$
$38744591877656925789747974914319422889611373939731$
that breaks up into 10 reversible 10 digit primes in order
$$3139971973, 7866347113, 9144865157, 726948...
Lookie there for an explanation.
I'm quite amazed by its beauty.
It's more appealing than my current set of love-affairs.
Well, I see it from a simple example. Many ancient philosophers were also mathematicians. (Rene Descartes comes to mind.) I think you need philosophical insight in order to do mathematics. That's how I see them as relatives: One does not exist without the other.
I don't think you can find a mathematician who doesn't have an appreciation for philosophy; though you MAY find a philosopher who has no appreciation for mathematics.
what do you take to be philosophy? love of wisdom? i know mathematicians that don't care much about this kind of thing except in their narrow range of interests
mathematics is useful and beautiful and the truths you get are relatively "pure", but it doesn't really get at "what it means to be a man", "what is the ultimate nature of reality", etc, which are generally considered to be in the domain of philosophy
i think you are coming at this from what would be called a Gnostic point of view
Do you mean that mathematician despise "truth in general" outside of mathematics? I see mathematicians as heavily valuing general truths in mathematics.
It is true that we can take a mathematical idea and use it physically, but the idea itself still seems to be mental. Its manifestations in nature (if there are any), however, are physical.
physicists and mathematicians can get veyr poetical when set free: that does not mean they interact in that way with their subject matter and what they actually do
Limitless, have you been to a math seminar? It's about the concrete theorems on the board. The theorem and the proof. And the mental process that leads to it.
you can treat vector spaces as a tool and still believe it to be a heavenly tool
i understand you don't want to spread a stereotypical image of matematicians, and i agree that that stereotype is false in its details, but there is some truth to this being at least a minority view
you can't just dismiss it as mere philosophizing while throwing back a drink after doing math
a carpenter can believe he is carving platonic forms in wood or that he is discovering a new piece of furniture rather than creating it. that doesn't effect what he does in terms of output
Ah, so you're going for that allegory where one builder says "I'm building a wall" and another says "I'm building a cathedral" with respect to the same thing they're doing?
"For this reason, the good Christian should beware not only numerologists, but all those who make impious divinations, above all when they tell truth. Otherwise, they may deceive the soul, and ensnare her in a pact of friendship with demons."
if you have a smooth mapping from $f: M\to N$, $M$ and $N$ smooth manifolds, the preimage $f^{-1}(\text{constant})$ is an embedded manifold if the $df$ is nonzero at the constant?
@MarianoSuárezAlvarez i'm thinking of the simple example $\mathbb{S}^n$. This is the $f^{-1}(1)$ where $f(x)=|x|^2$. Lee says that because the derivative is nonzero away from the origin, all of the level sets $f^{-1}(\text{nonzero})$ are embedded submanifolds. The preimage of zero is not. So the point is that the differential map between the spaces "degenerates" at zero? that there is no nbhd of the origin in $\mathbb{R}^n$ such that the differential map has constant rank?
@MarianoSuárezAlvarez, i'm trying to do the first problem in Lee on regular values. I have $F:\mathbb{R}^4\to\mathbb{R}^2$, with $F(x,y,s,t)=(x^2+y,x^2+y^2+s^2+t^2+y)$. I want to show that $(0,1)$ is a regular value of $F$.
So i have to constraints, $x^2=-y$, etc, and taken the partial derivatives. now i want to show that they can't all be linearly dependent i think
@MarianoSuárezAlvarez the rank is the number of linearly independent rows there are, yes? what do you mean by compute it? subject to the constraints see what needs to happen for it to be linearly dependent?
i feel like i may be being stupid, but it seems i want to make a change of variable $w^2=x^2+y$, and then our manifold $M$ becomes the level set $(w^2,w^2+y^2+s^2+t^2)=(0,1)$
and this is exactly the level set defining $S^2$ in $R^4$
@MarianoSuárezAlvarez
err, my change of variables should be $w^2=x^2+y$
under this change of variables you just map w^2 to x^2
There is no Supreme Court for mathematical notation; there were no commandments handed down on Sinai concerning operational precedence; all there is, is convention, and different people are free to adhere to different conventions. Wise people will stick in enough parentheses to make it impossible...
it's like how people say "i could care less" -- you probably know what they mean, but others will be more pedantic about it. If you want to remove any ambiguity, remove the ambiguity.
@MarianoSuárezAlvarez i'm doing a problem in Lee that i think is nearly trivial, i was hoping you could evaluate my opinion. for $c$ irrational, let $\gamma(t):\mathbb{R}\to\mathbb{T}^2$ be defined $f(t)=(e^{2\pi i t},e^{2\pi i c t})$. I want to show that $\gamma(\mathbb{R})$ is NOT an embedded submanifold. but it was previously proved that $\gamma$ is not an embedding (since $\gamma(\mathbb{Z})$ has limit points)
and we have two theorems saying that embedded submanifolds are precisely the images of smooth embeddings
What about the reversible 100 digit prime (i.e. it is a prime if written backwards)
$31399719737866347113914486515772694858917594191229$
$38744591877656925789747974914319422889611373939731$
that breaks up into 10 reversible 10 digit primes in order
$$3139971973, 7866347113, 9144865157, 726948...
