Do you believe in god or math?

[The Grand Nudger]

While your current line of convo isn't what I was referring to, in all likelihood, yeah, whatever you're about to say has a high probability of lunacy.

[Abracadabra]

While your current line of convo isn't what I was referring to, in all likelihood, yeah, whatever you're about to say has a high probability of lunacy.

You say that simply because I have an open mind toward the possibility that there may be a spiritual essence to reality?

[The Grand Nudger]

No, because you attempted to make 1+1=3. Such a gross misunderstanding of that fundamental of a concept isn't likely to lead to a sage-like understanding of any concept which is built upon the foundations of math.

[CliveStaples]

Ok, well let's talk just a little bit about set theory and see if we have even remotely the same idea of what it even emtails.

I would like to ask you the following questions just to get an idea of where you're coming from.

1. When do you believe that set theory first became a rigorous mathematical concept.

Hmm. Probably the work of Zermelo, Franco, and Russell in the early 20th century.

2. Who, if anyone, do you associate with introducing this concept formally to the mathematical community. (Feel free to name multiple mathematicians if you like)

Hmm. That's a tough question. I'd probably say Cantor, Weierstrass, or Dedekind.

3. Do you believe that the basic idea of a set was a required intuitive concept prior to this formalization?

Probably not. Greeks did number theory without really thinking of sets, so far as I know.

4. What is the formal mathematical definition of number in general?

If you believe there is more than one formal definition of number please explain as many as you can. (briefly) You can assume a simple cardinal definition here of the "Natural Numbers" or possible include a definition of Real Numbers if you like. You don't need to go off in la la land, speaking to the issues of imaginary numbers, complex numbers, irrational numbers, and so on. Let's try to keep this relatively basic. No need to make a essay out of it. I'm not going to pick at it. I'm just curious of how you think of these basic concepts.

It depends. You can define them algebraically, seeing a "number" as being part of a structure. Weierstrass defined the real numbers in terms of convergent sequences of rational numbers, which were defined in terms of integers, which were defined in terms of natural numbers, which were sort of assumed to exist.


I tend to think the algebraic definition makes the most sense. You don't really care about the numbers themselves, but about how they relate to each other (due to the operations defined on them as a set).

5. What do you believe is the formal definition of the Number One?
(feel free to give more than one definition if you like)

1 is the multiplicative identity (i.e., for any number x, x*1 = 1*x = x), 1 is not equal to 0, and 1 = s(0) (1 is the successor of 0 according to the definition of natural numbers from the Peano axioms).

6. How would you define a "Set"?

A set is a collection of objects that has a well-defined rule for inclusion in the sense that for all x and any set S, either x is in S or x is not in S but not both.

[Abracadabra]

No, because you attempted to make 1+1=3. Such a gross misunderstanding of that fundamental of a concept isn't likely to lead to a sage-like understanding of any concept which is built upon the foundations of math.

I thought I had already explained to you that I did that specifically to demonstrate just how important the units of quantity are.

1+1=2 is a meaningless concept in shorthand notation if you don't fully understand what these things mean.

You can't apply mathematics to the physical world and expect it to work if you don't pay very close attention to making sure that the "oneness" of the objects that you are quantifying is "well-defined'.

~~~~

I mean, gee whiz, we are light years away from the concepts and points that I would truly like to be discussing. We can't even get off the ground at all here.

You haven't yet understood this very simple point that I was making.

Neither has CliveStaples. He's also expressing a loss of understanding of what I mean by the need for things to be quantitatively well-defined.

He's totally lost in the pure abstraction of axiomatic thinking.

Hey, I fully understand how axiomatic systems works. Euclid gave us that gift of the axiomatic approach and mathematicians have run with it to extremes. And it certainly has panned out and is useful within certain restraints.

However, ironically it is this axiomatic approach to the formalism that is the the very thing that is crippling it.

Well, that's not exactly true. The ideas that I would like to propose to change things up can be defined axiomatically too. So it's not the axiomatic approach itself that is the problem. It's that this approach has caused certain things to become invisible.

So the very first thing that needs to be done is to back up and get past the axiomatic approach long enough to see what is currently invisible. Then after we see the problem, we can return to a the axiomatic approach again armed with this new insight.

So my example of how I could design an equation that ends up being 1+1=3 in shorthand notation was not an attempt to prove that mathematics can easily fail. That would be utterly stupid, because as you point out I broke the rules of "applied" mathematics.

However, it still served to demonstrate the importance of physical units of "oneness".

And that was the POINT I was attempting to expose.

You've got to comprehend very clearly the importance of the physical meaning of a quantity of "ONE" before it's even worth moving on to the next step.

Moving forward without that comprehension being made crystal clear would be totally futile.

And to be perfectly honest I don't see anyone grasping this trivial first step.

Rythmn thinks it was a cheap shot to belittle mathematics that miserably failed.

CliveStaples made it clear that he doesn't yet understand what I'm even talking about when I say "well-defined" quantitative property or "oneness".

So even if you guys were taking me seriously we appear to be at a dead end already anyway.

[Categories+Sheaves]

1. We've had primitive notions of what it means to have a collection of stuff, but the formalism I'm interested in started with Cantor. Some kinks in Cantor's business had to get smoothed out, and then we got the Zermelo-Fraenkel business we have today. So by that, rigorous treatments of sets started at least before 1880. But the foundations didn't get a rigorous treatment until about 1900-1920.

2. Cantor sent out the party invitations. Russell, Zermelo & Fraenkel kept the party alive after Cantor vomited and passed out.

3. Nobody invents some sort of mathematics without any notion of what it could be about. Intuition precedes the praxis, the praxis precedes the formalism.

4. Numbers started out as the formalization we used to describe our concepts of quantity. So the collection of our numbers is a list of names There is a strict ordering on these names, and there is a binary operation on these names (addition) that plays well with the ordering.

But there's a large family of groups, semigroups, rings, fields, and other things that extend that basic intuition in different direction. Nearly all of these things are of some interest. But some of them (e.g. p-adic integers) may not jive with the original intuition. To each their own.

5. The multiplicative identity if we've defined multiplication. The generator of our semigroup if we're in that world. The positive generator if we're in a group. It's the same 'flavor' of thing in almost all situations, but it still depends on what sort of math-world we're working in.

6. 'Set' is one of those primitive concepts. If I want to talk about some things, I have to find some way of pointing to the things I'm talking about. I have to... sort of wrap them up in brackets, hold them up and say, "these things, right here." So I can't describe what a set is without appealing to my audience's notions of what a set is. The axioms don't provide a definition of a set. They're a framework for constructing and manipulating sets.

[The Grand Nudger]

"My lofty concepts are misunderstood, Rhythm doesn't get it, Clive doesn't know what I'm talking about." That's fine, demonstrate the veracity of this statement at any time you feel is convenient. Show us what we don't get, explain what is not understood.

[CliveStaples]

However, it still served to demonstrate the importance of physical units of "oneness".

And that was the POINT I was attempting to expose.

You've got to comprehend very clearly the importance of the physical meaning of a quantity of "ONE" before it's even worth moving on to the next step.

Moving forward without that comprehension being made crystal clear would be totally futile.

And to be perfectly honest I don't see anyone grasping this trivial first step.

Rythmn thinks it was a cheap shot to belittle mathematics that miserably failed.

CliveStaples made it clear that he doesn't yet understand what I'm even talking about when I say "well-defined" quantitative property or "oneness".

So even if you guys were taking me seriously we appear to be at a dead end already anyway.

I don't see a distinction between "oneness" and "the image of 0 under the successor function" where the successor function is defined in accordance with the Peano axioms. They're the same thing. You start with an additive identity; the "next" thing you have is 1, or "oneness", etc.

[Categories+Sheaves]

You've got to comprehend very clearly the importance of the physical meaning of a quantity of "ONE" before it's even worth moving on to the next step.

Moving forward without that comprehension being made crystal clear would be totally futile.

And to be perfectly honest I don't see anyone grasping this trivial first step.

I do not need a crystal clear comprehension of my hammer before I strike a nail sitting in front of me. I sense that the head and handle are sturdy and firmly wedged together, and that is enough to proceed with my business.

We have a primitive notion of what I could be referring to when I say "this sort of thing," whatever that thing is. Whenever we wish, we can have our "one" become a stand-in for copies of this thing. If I see multiple two-by-four planks of wood, I can count them and say, "I have this many two-by four planks of wood". If I have a single plank and you split it lengthwise, it is no longer a two-by-four. This is not by a fault in our numbers, it is simply a statement that the wood in front of me no longer fits my definition of "two-by-four plank*".

Math is obviously not God. Math is also not physics. Math is the art of knowing how to apply some simple problem-solving techniques, and occasionally the art of good notation. Nothing more. To whatever extent we see multiple things that are functionally equivalent to us, we will have a use for this concept of counting. To whatever extent we try to group or categorize things, we will have a use for this concept of sets.

As CliveStaples was pushing me on earlier, the strength of these things is in their generality. I don't have to restrict myself to the nails I imagined at the time I made my hammer, I can hammer other things that behave similarly to nails. How can I tell whether an object can be hammered into something else? I ought to be able to list all the properties of nails and wood that allowed my hammer to work on them... how about we write these things out and call them axioms?

Also: when do we get to your thoughts about contemporary math and stuff like that?

*And yes, I know the dimensions of the base of these planks aren't exactly two by four inches

[Categories+Sheaves]

Hey, I fully understand how axiomatic systems works. Euclid gave us that gift of the axiomatic approach and mathematicians have run with it to extremes. And it certainly has panned out and is useful within certain restraints.


However, ironically it is this axiomatic approach to the formalism that is the the very thing that is crippling it.

Well, that's not exactly true. The ideas that I would like to propose to change things up can be defined axiomatically too. So it's not the axiomatic approach itself that is the problem. It's that this approach has caused certain things to become invisible.

So the very first thing that needs to be done is to back up and get past the axiomatic approach long enough to see what is currently invisible. Then after we see the problem, we can return to a the axiomatic approach again armed with this new insight.

ooohhhhhhh.... wow. I'm stupid and can't read.

Even if you don't feel like discussing those ideas anymore, could you drop some links/books/etc. that demonstrate them well?