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August 19th, 2016, 03:06 PM   #221
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It turns out that I had been labouring under a misunderstanding.
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Originally Posted by Maschke View Post
Ye Gods is a pretty minor reaction in the scheme of things... I deleted worse yesterday.
I'd be grateful if you confine yourself to explaining where you think I've made an error. There's no need for outbursts of that nature.
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August 19th, 2016, 03:21 PM   #222
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It turns out that I had been labouring under a misunderstanding.

I'd be grateful if you confine yourself to explaining where you think I've made an error. There's no need for outbursts of that nature.
I'm not going to get into a dispute with you nor am I going to retract what I wrote.

Ye Gods is like a humorous slap to the forehead. It's playful. Yes it's mildly chiding. Well deserved in my opinion and in no way beyond the limits of good taste. You are one of the people here who plays someone knowledgeable; so your errors have a wider effect than they might otherwise have. So yes I call out your errors when I see them. If you want less chiding, make fewer mathematical errors. How's that?

Last edited by Maschke; August 19th, 2016 at 03:33 PM.
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August 19th, 2016, 04:16 PM   #223
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Okay, good point well made. And I obviously agree with it because it's kind of confirmation of what I said in my prior post!
If I've helped move you from a momentary flirtation with constructivism, to the permissive view of pure existence; then I'm gratified.

Just as we recently abstracted the concept of order from the concept of set; we have now abstracted existence from constructibility. When we say a thing exists; we mean ONLY that it exists. We don't claim that we can build one, or describe one, or even comprehend one; only that it exists.

Once you accept that, there are so many cool thing you can do. Like all of modern math. But without pure existence, if you had to algorithmically construct everything, you just be slogging through one technical problem after another and you'd never get any math done.

You are right: It's just for convenience. Math is not dogmatic. Math is pragmatic.

If people understood that, math would be totally demystified.

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I was thinking in the shed (like you do)
I'm sorry, is that a Britishism? I'm in the colonies. Does it mean what I think it does?

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that the whole point about axiomatic systems is that they provide a structure - a set of ground rules - that we can use to build the things that we want to build.
Yes exactly. We did math for thousands of years without foundations. In the 19th century we realized we needed foundations. Out of that came set theory, which took over 20th century math. It's arguably already on the way out. Next century, who knows.

Math is a historically contingent activity of flawed human beings.

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It seems reasonably clear that we can build useful sets of axioms and we can build not-so-useful sets of axioms. It's very hard to justify the usefulness or otherwise of any system without a full understanding of all the problems and circumstances we may encounter. We are not that omniscient.
It took about 50 years from Cantor's first paper to the axioms of set theory in their modern form. And research continues on new and better axioms. Nobody thinks the current axioms are the last word.


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Regardless, I guess my question would be - in what circumstances are sets of greater cardinality than the reals useful?
That's a technical question that has an answer. The set of functions from the reals to the reals has the cardinality of the powerset of the reals. That or the next powerset up is pretty much all the sets used by mainstream mathematicians. Set theoriest study far larger cardinalities, but mainstream math is basically all inside the powerset of the powerset of the reals or less.

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Do they only (to our current knowledge) solve "arcane" mathematical issues or do they allow us to solve actual real problems in the real world. "real" is possibly an unfortunate word.
Large cardinals (cardinals that are so large they can't even be proved to exist in standard set theory) have implications in mathematical logic, which relates to theoretical computer science at some level. Obviously nobody's building bridges with this stuff but you just don't know what the next few decades will bring.

Historically, the arcane and useless math of one century is the applied math of the next. Non-Euclidean geometry was a strange curiousity in the 1840s, then Einstein came along and showed that it was the geometry of the world.

You never know when useless math will become useful.


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I'm curious that's all. I'm starting to accept pretty much everything as given
You have been assimilated

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because we have the solid caveat that - "this is the system that we chose to work in because we find it useful" and "there are other systems but they have their own problems and are less useful".

Which is fine and as it should be.
Yes exactly. Pragmatics. We invent the math we need to solve whatever problems we have. Our very concept of what math is keeps changing. You just learn to accept math on its own terms.

I should mention that some critics think math has gotten dangerously out of touch with reality in the past century. I believe a lot of physicists have that opinion. I wouldn't want to not mention that point.
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August 19th, 2016, 04:36 PM   #224
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I'd be grateful if you confine yourself to explaining where you think I've made an error. There's no need for outbursts of that nature.
ps -- Ok now I feel guilty. I would not call my post an "outburst." But I'll respect the fact that you regard it so.

In the future I will be more circumspect in my responses to your posts.
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August 20th, 2016, 04:20 AM   #225
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If I've helped move you from a momentary flirtation with constructivism, to the permissive view of pure existence; then I'm gratified.
I think that it's valuable to see both sides of the argument. It's almost like the "pure" side and then the "err, can we actually do something" side. Both have merit.

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Once you accept that, there are so many cool thing you can do.
And now I should start to explore them.

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Originally Posted by Maschke View Post
If people understood that, math would be totally demystified.
Certainly if I'd realised it 30 years ago I might have gotten a better degree!

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Originally Posted by Maschke View Post
I'm sorry, is that a Britishism? I'm in the colonies. Does it mean what I think it does?
LOL. No. The shed is traditional male bolt-hole in the UK. I guess being superceded by the "man-cave". I was actually sitting in my shed drinking beer!

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The set of functions from the reals to the reals has the cardinality of the powerset of the reals.
That makes sense. Kind of feels like it should be true.

I was wondering about infinite sequences of reals and things like that.

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You have been assimilated
Noooooooo!!!!!!!!

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I believe a lot of physicists have that opinion. I wouldn't want to not mention that point.
Well, if you consider that in Quantum mechanics we build constructor and destructor functions that enable us to move up and down quantum states, and consider that everything has a quantum state. Maybe the universe is just a natural number and the multiverse is a real number.

Who'da Thunk?

<runs from keyboard laughing maniacally to the sound of thousands of heads slapping into desks...>
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