Representations, Wittgenstein, and all the rest

[Vanadium 50]

I spun off the entirety of off-topic posts from ...
https://www.physicsforums.com/threads/representations-of-group-z_2.997719/
..., hope I didn't make a mistake, and thus created a more or less general thread for free associations, so that we can continue this small talk about mathematics without derailing the original thread.

___

Your answers till now were not helpful.

Blaming people for "unhelpful answers" when you have an unclear question does not put the blame in the right place.

Either your two 1's are distinct or they are not. If they are not distinct, you shouldn't have two of them. If they are, you shouldn't use the same symbol for them.

 

[sysprog]

Blaming people for "unhelpful answers" when you have an unclear question does not put the blame in the right place.

Either your two 1's are distinct or they are not. If they are not distinct, you shouldn't have two of them. If they are, you shouldn't use the same symbol for them.

For what it may be worth, I agree with you here.

From http://wittgensteinrepository.org/agora-alws/article/view/2781/3286

Abstract
According to Wittgenstein, self-identity is not a genuine relation that an object bears to itself. That is, it is not trivially and universally true that an object is identical to itself. This paper will employ the distinction between nonsense and senselessness and explain why, according to Wittgenstein, assertions of self-identity are nonsense. This implies that assertions of self-identity cannot be used to refer to objects qua objects, and this has philosophical consequences for, inter alia, Russell’s Axiom of Infinity and Frege’s derivation of the natural numbers.​

 

[fresh_42]

I am so tempted to continue the debate about Wittgenstein's concept of self-identity. From a mathematical point of view, does this mean equivalence instead of identity? I remember a tough debate about whether $\dfrac{1}{2}=\dfrac{2}{4}$ or $\dfrac{1}{2}\sim \dfrac{2}{4}.$ My position was the latter. Did Wittgenstein go even further and claims $1\sim 1$ instead of $1=1$?

 

[sysprog]

@fresh_42 , a little further on, Waismann is quoted:

If it makes sense to ask whether the [two] armchairs can be distinguished, then they are two armchairs; if this question makes no sense, then it is one chair. In other words, the question whether two things are identical is not the question whether they can be distinguished, but whether it makes sense to ask whether they can be distinguished. (Waismann 1977, 26)​

At the risk of going off-topic, albeit yet sticking to Physics, let's please look ahead a bit:
what do you think about the applicability of the enumeration of the non-commutative SU(3) homology group to the enumeration of the quark set in quantum chromodynamics?

 

[Vanadium 50]

Is 1 = 1? Is A = A? Is helping futile?

 

[sysprog]

I am so tempted to continue the debate about Wittgenstein's concept of self-identity. From a mathematical point of view, does this mean equivalence instead of identity? I remember a tough debate about whether $\dfrac{1}{2}=\dfrac{2}{4}$ or $\dfrac{1}{2}\sim \dfrac{2}{4}.$ My position was the latter. Did Wittgenstein go even further and claims $1\sim 1$ instead of $1=1$?

I take the position that quantitatively or quantificationally, $\frac 1 2$ and $\frac 2 4$ are obviously absolutely identical; however, they are, equally obviously and absolutely, notationally different $-$ computationally, storing one half is not always or necessarily the same as storing two fourths, and that could matter if someone cares about the difference.

In his On Certainty Wittgenstein contradicted some things that he'd said in his Tractatus Logico-Philosophicus (the title of which was a play on Spinoza's Tractatus Theologico-Politicus); in particular, he apparently changed his mind about conventions regarding the naming of objects, and made reference in related contexts to what he decided to call 'language games'.

I still like what he did in his Tractatus $-$ the orderly numbering of statements and substatements gives the work an especial appeal for a programmer $-$ and the last statement, number 7, "wovon man nicht sprechen kann, darueber muss man schweigen." (of that of which one cannot speak, one must be silent) is in my opinion beautifully ironic.

 

[fresh_42]

I take the position that quantitatively or quantificationally, 12 and 24 are obviously absolutely identical; however, they are, equally obviously and absolutely, notationally different − computationally, storing one half is not always or necessarily the same as storing two fourths, and that could matter if someone cares about the difference.

My favorite comparison is a piece of cake: it makes a difference whether you buy one half or two quarters. The pieces are obviously not the same. In our example where $1$ could have been the identity element of various groups, it is important to understand that the property of being the identity element is not context free, an environment is necessary.

Guess at least half of philosophy and even more Wittgenstein's is about that property of languages: To which extent does a name of an object identify the object? And this is also important in programming in general, where general means the entire product, not only the code. Role, individual, and task are three different things, albeit often identified. If so, then at least it guarantees follow-up orders.

 

[Vanadium 50]

Far be it for me to remind the Mentors of our rules on philosophy, but I will say if you need to invoke Wittgenstein to try and figure out what the OP means, it might be a sign that there might be a little bit of a clarity shortage.

 

[sysprog]

Hey, please wait, @fresh_42 $-$ apparently the board software is causing you to misquote me $-$ I didn't say that 12=24 I said something that means the obvious fact that one half is in quantity equal to two quarters.

 

[sysprog]

To which extent does a name of an object identify the object?

Many times a PER (programming error) has turned out to be yet another error such that a LOAD instruction should have been a LOAD ADDRESS instruction, or vice versa.

 

[sysprog]

My favorite comparison is a piece of cake: it makes a difference whether you buy one half or two quarters. The pieces are obviously not the same.

That's true $-$ the two quarters have a greater surface area, and so dry out or ferment faster.

 

[Mark44]

I am so tempted to continue the debate about Wittgenstein's concept of self-identity. From a mathematical point of view, does this mean equivalence instead of identity? I remember a tough debate about whether $\dfrac{1}{2}=\dfrac{2}{4}$ or $\dfrac{1}{2}\sim \dfrac{2}{4}.$ My position was the latter.

My position is the former; i.e., that there is no numeric difference between $\frac 1 2$ and $\frac 2 4$. Obviously, one can cook up an example with half a cake vs. two quarters of a cake, but if we look at the fractions as pure numbers, they are equal.

And certainly, the number 1 is equal to itself, philosophical sophistry notwithstanding.

 

[fresh_42]

My position is the former; i.e., that there is no numeric difference between $\frac 1 2$ and $\frac 2 4$. Obviously, one can cook up an example with half a cake vs. two quarters of a cake, but if we look at the fractions as pure numbers, they are equal.

I know, I remember it. In the end it is a rather academic discussion, maybe pragmatism versus formal algebra: https://ncatlab.org/nlab/show/field+of+fractions

 

[sysprog]

May all of us computer nerds please unite regarding these superextremely obvious two things: $\frac 1 2 = (2 \times \frac 1 4)$ and $1=1$.

 

[fresh_42]

A representation of a group is just a group homomorphism to the general linear group. Nothing requires it to be injective, that is in fact what a faithful representation is.

That was my mistake in post #2, too. We are so used to it on a physics website that it is already our second nature, but general linear is not required. Any automorphism group will do.

 

[Office_Shredder]

That was my mistake in post #2, too. We are so used to it on a physics website that it is already our second nature, but general linear is not required. Any automorphism group will do.

I think general linear group is the usual mathematical definition for pure mathematicians, and is what wikipedia uses as the definition.

It doesn't matter that much for this thread's question though, so let's not muddy it up. We can discuss in another thread if people care.

 

[fresh_42]

I think general linear group is the usual mathematical definition for pure mathematicians, and is what wikipedia uses as the definition.

It doesn't matter that much for this thread's question though, so let's not muddy it up. We can discuss in another thread if people care.

This thread has already derailed, and the OP didn't show up to clarify his question anyway.

I like automorphism groups over general linear, as it means that I do not have to make an artificial distinction between operation and representation.

 

[sysprog]

@fresh_42 maybe you care that a so-called spherical cow is, based on the realities of its alimentarities, topologically homeomorphic to a toroid rather than to a sphere.

 

[fresh_42]

@fresh_42 maybe you care that a so-called spherical cow is, based on the realities of its alimentarities, topologically homeomorphic to a toroid rather than to a sphere.

Nope, because the next question would be about the genus, and that is definitely not an easy question!

 

[sysprog]

Nope, because the next question would be about the genus, and that is definitely not an easy question!

Isn't the genus of a torus simply 1? ok maybe it's not so simple in the case of a cud-chewing spherico-toroidal ox (aurochs $-$ bovine) with 7 stomachs.

Professor, I honestly believe that 'not an easy question' is a term of endearment for you − also I think that you may be genuinely on to something regarding toroidal genus − maybe there is a convergence in proteasome activity in a direction of emergence of quaternary structures − after that it's just icky − apicobasal abluminal polarity − oops, maybe that belongs in the bio areas $\dots$

Even with compromise on specifics, the spherico-toroidal cow still has genus one among its genera, right?

 

[fresh_42]

I spun off the entirety of off-topic posts from ...
https://www.physicsforums.com/threads/representations-of-group-z_2.997719/
..., hope I didn't make a mistake, and thus created a more or less general thread for free associations, so that we can continue this small talk about mathematics without derailing the original thread.

Isn't the genus of a torus simply 1? ok maybe it's not so simple in the case of a cud-chewing spherico-toroidal ox (aurochs $-$ bovine) with 7 stomachs.

Not only the stomach. The genus of mammals is already difficult to define: the Eustachian tube, the eyes, and last but not least the food channel. How do we handle the drinking channel?

Professor, I honestly believe that 'not an easy question' is a term of endearment for you − also I think that you may be genuinely on to something regarding toroidal genus − maybe there is a convergence in proteasome activity in a direction of emergence of quaternary structures − after that it's just icky − apicobasal abluminal polarity − ...

Uhmm, what?

...oops, maybe that belongs in the bio areas $\dots$

Only until they gave as a definition. After that it's a matter for topologists, maybe even knot theorists!

Do mammals have a Jones polynomial, and if so, which one?

 

[sysprog]

Do mammals have a Jones polynomial, and if so, which one?

I'm pretty sure that I don't know as much knot theory as an average boy scout or girl scout $-$ two half-hitches and an ovine bend (sheep bend) and I get confused $-$ I think that I have missed a cleverness in your question; however, if it's not a hairless mammal, there may be possibilities for tuple braiding. If we don't sweat the small stuff $-$ all that ductwork $-$ the spherical cow continues to be invariantly homeomorphic to a toroid.

 

[Stephen Tashi]

The way this thread is displayed, it looks like it was split-off from a different thread. Is that the case?

Taking the apparent topic seriously, we can ask if there are any logical pitfalls in defining "identical to" as a property using a statement universally quantifier over equivalence relations. Let $E(x,y)$ denote an equivalence relation $E$ between $x$ and $y$ that describes them being "the same" with respect to some aspect. Do we get into trouble by defining "identicallity" $I(x,y)$ to mean $\forall E: E(x,y)$ ? Do we run into something like Russell's paradox? There seems to be trouble in that an given equivalence relation like "is the same color as" is not defined for things such as real numbers.

When it comes to question like "1/3" vs "2/6", the issue of "the symbol for the thing" vs "the thing that is symbolized" arises. When a person says that "1/3" is "identical" to "2/6", presumably the claim is about the thing symbolized. However, all that can be written about a thing employs symbols for the thing. We could also say that all that can be thought about a thing involves, in some way, symbols or representations for the thing. e.g. When we think of an elephant we do not literally have an elephant inside our brain.

 

[Mark44]

ovine bend (sheep bend)

You've combined two knots into one -- the sheet bend and the sheep shank. If I remember right, a sheet bend is used for connecting two ropes of different diameters, and a sheep shank is used to shorten a rope.

 

[Mark44]

When it comes to question like "1/3" vs "2/6", the issue of "the symbol for the thing" vs "the thing that is symbolized" arises. When a person says that "1/3" is "identical" to "2/6", presumably the claim is about the thing symbolized.

In my earlier post I made an effort to qualify what I was saying; i.e., that 1/2 and 2/4 are numerically equal. It should be clear that I meant that the things were identical rather than the symbols.

 

[sysprog]

You've combined two knots into one -- the sheet bend and the sheep shank. If I remember right, a sheet bend is used for connecting two ropes of different diameters, and a sheep shank is used to shorten a rope.

You're right about that $-$ I conflated those two terms $-$ I guess that I know even less of knot theory than I previously supposed that I did.

 

[sysprog]

Not only the stomach. The genus of mammals is already difficult to define: the Eustachian tube, the eyes, and last but not least the food channel. How do we handle the drinking channel?

In the early '80s I had a fridge magnet that said "Beer is Food" $-$ my girlfriend didn't much like it, but she smiled about it a little.

 

[fresh_42]

In the early '80s I had a fridge magnet that said "Beer is Food" $-$ my girlfriend didn't much like it, but she smiled about it a little.

The little I eat I can drink as well.

Edit: It is even not yet finally decided, whether humans first used wheat to brew beer rather than to bake bread. Chances are high because of the kind of yeast they have found and the fact that beer is anti bacterial in contrast to bread.

 

[sysprog]

There seems to be trouble in that an given equivalence relation like "is the same color as" is not defined for things such as real numbers.

What about 'is the same number as' $-$ isn't that what we ordinarily mean when we say that a = b?

 

[Stephen Tashi]

What about 'is the same number as' $-$ isn't that what we ordinarily mean when we say that a = b?

That approach works if we are not trying to define the relation "is identical to" in a global manner and are content to define "is identical too" differently for different types of things.

The problem of defining "is identical too" is, of course, connected to the problem of defining "is different than". In mathematics, I suppose we don't have to worry about having a different definition for "is identical to" for different mathematical objects. However, in thinking about Nature, we conceive of it as consisting of unique events. Science relies on repeatable experiments. To be "repeatable", the experiments must be "the same" in some respects but they must also be unique and different in other respects - otherwise they would not be series of different experiments, but only a single experiment, performed once.

We have the idea that two people can talk about "the same" experiment in the strongest sense of "sameness" - i.e. in the same sense that they can speak of "the same" coffee cup or the "the same" person. In that context, whatever symbols or language they are using to symbolize the thing in question, they are both referring to the "same" thing, not to two different things that are the same in many aspects. This concept of the (general) relation of "sameness" seems to exist independently of any mathematical definition of what the thing in question is. But perhaps the concept can't be treated rigorously.