Algebraic Topology in the tv show The Big Bang Theory

[tylerfarzam]

in the tv show "The Big Bang Theory", Sheldon wrote a book called "A proof the algebraic topology can never have a non self-contradictory set of abelian groups". Is this just a random set of words that is meant to sound smart but in reality means nothing or is it accurate? If it is, what does it mean. I looked it up and someone said it was the first option but they didn't explain why.

 

[WWGD]

I have never heard of anything remotely Mathematically reasonable that reads like that statement; specifically , the expression " Non-contradictory Abelian group" has no meaning I am aware of.

 

[tylerfarzam]

I have never heard of anything remotely Mathematically reasonable that reads like that statement; specifically , the expression " Non-contradictory Abelian group" has no meaning I am aware of.

okay thanks

 

[PeroK]

in the tv show "The Big Bang Theory", Sheldon wrote a book called "A proof the algebraic topology can never have a non self-contradictory set of abelian groups". Is this just a random set of words that is meant to sound smart but in reality means nothing or is it accurate? If it is, what does it mean. I looked it up and someone said it was the first option but they didn't explain why.

All the maths and physics in that show is authentic, so I was surprised by this. I haven't seen that episode.

From what I found on line, he wrote that paper when he was 5 years old. And, the joke is that as an adult he cannot accept that even as a five-year-old he could have been wrong about anything.

 

[tylerfarzam]

All the maths and physics in that show is authentic, so I was surprised by this. I haven't seen that episode.

From what I found on line, he wrote that paper when he was 5 years old. And, the joke is that as an adult he cannot accept that even as a five-year-old he could have been wrong about anything.

I can see where your coming from but the question is not wether he was right or not, but if that is just a collection of words meant to sound "smart" or not. So I was asking if that collection of words make sense and if there is even such a thing as "self- contradictory abelian groups in algebraic topology".

 

[fresh_42]

if that is just a collection of words meant to sound "smart" or not

Yes.

As already mentioned: "non self-contradictory set of abelian groups" is rubbish. A set cannot be "contradictory" and "algebraic topology" is just a branch of mathematics which probably sounds sophisticated in the ears of common people. Well, it actually can be sophisticated and Abelian groups play a role in there, but not as a subject of investigation, only as a quality of some groups there.

 

[tylerfarzam]

Yes.

As already mentioned: "non self-contradictory set of abelian groups" is rubbish. A set cannot be "contradictory" and "algebraic topology" is just a branch of mathematics which probably sounds sophisticated in the ears of common people. Well, it actually can be sophisticated and Abelian groups play a role in there, but not as a subject of investigation, only as a quality of some groups there.

Okay thank you. That is all I was asking.

 

[Rohan]

Quite likely:
"A proof that algebraic topology can never have a non self-contradictory set of Abelian groups".
is rubbish; but shouldn't we start from the term , " set of Abelian groups". ?
(I have recently got up to here in my attempt to understand modern mathematics
so as a beginner please correct me if I am not understanding !)
- A group is just a set, for which an operation eg * is defined (I like to think of the multiplication of square matrices !)
-ordinary groups have the properties of (a)closure (b) associativity (c)identity (d)inversion and
-an "Abelian group" is just a group with the additional property of (e) commutation ( eg A x B = B x A for square matrices)
Thus " a set of Abelian groups", could mean a set of these, now sets can famously contain all kinds of things.
I seem to recall one as part of a famous paradox which " was the set of all sets , which do not contain themselves";
which was described as a "contradictory set" ?
Thus couldn't the statement "that algebraic topology can never have a non self-contradictory set of Abelian groups";
be interpreted as stating ' all sets of Abellian groups in algebaic topology are contradictory ?'
Of course what this means and whether it is true are well beyond my ability !

 

[fresh_42]

Quite likely:
"A proof that algebraic topology can never have a non self-contradictory set of Abelian groups".
is rubbish; but shouldn't we start from the term , " set of Abelian groups". ?
(I have recently got up to here in my attempt to understand modern mathematics
so as a beginner please correct me if I am not understanding !)
- A group is just a set, for which an operation eg * is defined

Yes, where this binary operation obeys certain rules.

(I like to think of the multiplication of square matrices !)

Those do not obey these rules in general. Better think of the multiplication of non zero numbers, or invertible square matrices.

-ordinary groups have the properties of (a)closure (b) associativity (c)identity (d)inversion and

Yes, that is one way to define it.

-an "Abelian group" is just a group with the additional property of (e) commutation ( eg A x B = B x A for square matrices)

Again, not in general. Matrix multiplication is "far more often" not commutative, as it is commutative.

Thus " a set of Abelian groups", could mean a set of these, now sets can famously contain all kinds of things.

Almost all.

I seem to recall one as part of a famous paradox which " was the set of all sets , which do not contain themselves";
which was described as a "contradictory set" ?

This certain set is excluded for a set if properly defined. Thus, and even for pure linguistic reasons, the term "contradictory set" doesn't make sense. It simply does not say in what the contradiction lies. E.g. we can talk about self-contradiction of a system, i.e. a set of rules which when applied create a contradiction. This demonstrates the next reason why "contradictory set" is nonsense: A contradiction means a conclusion leads to $true = false$, and a set per se cannot do this. Only its elements, so they are conclusion laws can do this. Abelian groups are no rules, and I've never heard of a binary operation on rules, let alone a commutative. One might be able to construct such a thing, but even then, the specification "Abelian groups" is far too general. And on top: neither of these things has anything to do with algebraic topology.

Thus couldn't the statement "that algebraic topology can never have a non self-contradictory set of Abelian groups";
be interpreted as stating ' all sets of Abellian groups in algebaic topology are contradictory ?'

No, as contradictory to what is still missing. An abelian group cannot be "contradictory" regardless how many you sample in a set.

Of course what this means and whether it is true are well beyond my ability !

The only sense why this quote has found its way into the show is, that all words in it can be looked up on Wikipedia and most people won't understand a single one of them. Fact-check-proof isn't intended in a tv-show.

 

[jbriggs444]

I seem to recall one as part of a famous paradox which " was the set of all sets , which do not contain themselves";
which was described as a "contradictory set" ?

That is Russell's Paradox. As understood now, it is not a contradictory set, but a contradictory description of a set. ZF set theory avoids the problem by not allowing "unrestricted comprehension". Unrestricted comprehension is the practice of defining a set by specifying a property without specifying what containing set elements with that property are to be drawn from.

 

[WWGD]

Quite likely:
"A proof that algebraic topology can never have a non self-contradictory set of Abelian groups".
is rubbish; but shouldn't we start from the term , " set of Abelian groups". ?
(I have recently got up to here in my attempt to understand modern mathematics
so as a beginner please correct me if I am not understanding !)
- A group is just a set, for which an operation eg * is defined (I like to think of the multiplication of square matrices !)
-ordinary groups have the properties of (a)closure (b) associativity (c)identity (d)inversion and
-an "Abelian group" is just a group with the additional property of (e) commutation ( eg A x B = B x A for square matrices)
Thus " a set of Abelian groups", could mean a set of these, now sets can famously contain all kinds of things.
I seem to recall one as part of a famous paradox which " was the set of all sets , which do not contain themselves";
which was described as a "contradictory set" ?
Thus couldn't the statement "that algebraic topology can never have a non self-contradictory set of Abelian groups";
be interpreted as stating ' all sets of Abellian groups in algebaic topology are contradictory ?'
Of course what this means and whether it is true are well beyond my ability !

As Fresh-heimer said, Non-Contradictory ( or , contradictory) is not a term that applies outside of the scope of Logic. Within Algebra or Algebraic Topology the term just does not apply, just like the term " Non-Blue" groups would not apply either, or " Non-Blue" legal terms, etc.

 

[WWGD]

That is Russell's Paradox. As understood now, it is not a contradictory set, but a contradictory description of a set. ZF set theory avoids the problem by not allowing "unrestricted comprehension". Unrestricted comprehension is the practice of defining a set by specifying a property without specifying what containing set elements with that property are to be drawn from.

Expanding

Quite likely:
"A proof that algebraic topology can never have a non self-contradictory set of Abelian groups".
is rubbish; but shouldn't we start from the term , " set of Abelian groups". ?
(I have recently got up to here in my attempt to understand modern mathematics
so as a beginner please correct me if I am not understanding !)
- A group is just a set, for which an operation eg * is defined (I like to think of the multiplication of square matrices !)
-ordinary groups have the properties of (a)closure (b) associativity (c)identity (d)inversion and
-an "Abelian group" is just a group with the additional property of (e) commutation ( eg A x B = B x A for square matrices)
Thus " a set of Abelian groups", could mean a set of these, now sets can famously contain all kinds of things.
I seem to recall one as part of a famous paradox which " was the set of all sets , which do not contain themselves";
which was described as a "contradictory set" ?
Thus couldn't the statement "that algebraic topology can never have a non self-contradictory set of Abelian groups";
be interpreted as stating ' all sets of Abellian groups in algebaic topology are contradictory ?'
Of course what this means and whether it is true are well beyond my ability !

The issue of the set of all sets, and whether it contains itself is a response to the issue on whether any well-defined condition defines a set. This was assumed in Naive set theory, but this condition of set of all sets shows that not every well-defined condition defines a set.