## Status of Cantor's 'Continuum Hypothesis'

Discussions concerned with knowledge of measurement, properties, and relations quantities, theoretical or applied.

### Re: Status of Cantor's 'Continuum Hypothesis'

Natural ChemE » November 12th, 2015, 2:27 am wrote:someguy1,

I'm curious about your outlook on what's interesting/used/etc. Basically I feel like you're dismissing the tools that we currently use in applied fields based on their obscurity in non-applied fields, which seems very strange to me.

Hypothetically, say that engineers were using mathematics effectively unreproducible in terms of the systems that you're talking about. Say that human progress is advanced by progress in these tools unknown to non-applied mathematicians. In this hypothetical case, would you be interested in such tools, or would their obscurity within non-applied fields continue to cause them to be disinteresting to you?

I've drawn the distinction among the formal hyperreals as defined in math, the informal infinitesimals used by physicists, and the non-infinitesimal mathematical theory of limits. I didn't say physics was wrong or bad in any way; on the contrary, I said it was perfectly normal because physicists are interested in studying the universe, and not so much in the fine points of logical rigor. Was that not a satisfactory description of the situation?

Where did I dismiss anything? On the contrary. I pointed out that physicists regard mathematicians as excessively devoted to logical purity at the expense of studying the world. Did I not say that? Did you not read it? Did you read it and think I said something else?

I specifically made the point that math sometimes leads physics; and that physics sometimes leads math. Or maybe I lost that point in some edit. I'm perfectly aware of it. Sometimes the nonrigorous tools of physics prove valuable. Newton's calculus was one such example. Normalization is another.

Where did I dismiss anything?
someguy1
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### Re: Status of Cantor's 'Continuum Hypothesis'

ps ... Didn't mean for that to come out too crabby sounding :-) I'm not understanding the question. Can you just point out where I dismissed the practical approach, I'd understand your question better.

If anything, I've tried to be openminded to whatever the future decides. Which approach is in favor today is historically contingent. A hundred years from now perhaps set theory will be seen as a terrible mistake, and everyone will be doing intuitionist type theory or something. There is always some intellectual doubt about what we're doing. Even math is fallible. That's the real shock of 1840. Math doesn't tell us what's true. Only what's possible.

I make no claim that anything's better than anything else. I hope I placed the hyperreals in a historical context. Today nobody cares about them. If tomorrow morning Professor Such-and-So in Tibet uses the hyperreals to prove that $P \ \neq \ NP$ then everyone will care.

That's how science works. You never know what will be important tomorrow. I really thought this had been one of my core themes, that you have to have an open mind because you never know what new scientific discoveries and paradigms are coming. If I've been dismissive of anything I would like to know so that I can correct my error.
someguy1
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Posts: 671
Joined: 08 Nov 2013

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