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July 8th, 2014, 02:07 PM   #1
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Are you skeptical of pure math?

I'm a logician not a mathematician but I want to eventually work on logicism. That is where you try to show that math is a subset of logic. Essentially what you would do is show that all the math symbols can be expressed in the standard logical symbols: v & = -> ~ and a few other symbols. I've never been persuaded of any proof in non-classical logic except for a few here and there that tried to prove very uncontroversial things. Classical logic on the other hand I find at least parts of it convincing and persuasive. Whenever a logician starts trying to prove something of import I always find huge gaps in their reasoning and am never convinced. I have a hunch that pure math is the same way. I'm not skeptical of calculus or any math that is used to verify experimental results in particle physics because when you're forced to use math to verify experiment or travel to the moon it can be tested. Pure math on the other hand cannot be tested unless you're using a computer that checks that you're obeying all your rules which is what I eventually want to do when I start working on logicism. Pure math can be tested in the sense that you convince other mathematicians but a whole community of mathematicians can be in error, just look at Godel's Theorems which everyone believes (most simply accepting it on authority) but are most likely false. (If you object to that then I'm not interested in debating that right now). I've got a computer program that employs about 45 axioms and I want to use those axioms to prove various math theorems. I trust someone who uses computer to show that their rules are being followed but I don't see any logicians doing that except for a few people working on automated theorem proving.

So my question is how convincing do you find proofs in pure math?
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July 8th, 2014, 06:55 PM   #2
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Originally Posted by bobsmith76 View Post
I'm a logician not a mathematician but I want to eventually work on logicism. That is where you try to show that math is a subset of logic. Essentially what you would do is show that all the math symbols can be expressed in the standard logical symbols: v & = -> ~ and a few other symbols.
Wasn't that done a hundred years ago?

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I've never been persuaded of any proof in non-classical logic except for a few here and there that tried to prove very uncontroversial things.
You can typically embed non-classical logics in a sufficiently powerful standard logic.

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Pure math on the other hand cannot be tested unless you're using a computer that checks that you're obeying all your rules which is what I eventually want to do when I start working on logicism.
There are plenty of extant systems that do that: Metamath, Mizar, Coq, etc.

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a whole community of mathematicians can be in error, just look at Godel's Theorems which everyone believes (most simply accepting it on authority) but are most likely false.
This pretty much shows that you're a crank. I was hoping this could be a reasonable and enlightening discussion, because I'm really interested in the topic, but I guess that was too much to hope for.

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So my question is how convincing do you find proofs in pure math?
Depends on what you mean. When I read carefully through a published proof, I can become entirely convinced of the truth of the proposition at hand. I may not trust that every step of the proof is correct or sufficient, but the parts I find unconvincing I am able to prove in some other way.
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July 8th, 2014, 09:55 PM   #3
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Wasn't that done a hundred years ago?

This pretty much shows that you're a crank. I was hoping this could be a reasonable and enlightening discussion, because I'm really interested in the topic, but I guess that was too much to hope for.
Right, anyone who denies an orthodoxy is a crank. Based on that logic, Galileo, Einstein, and numerous others would be cranks.

The only difference between a crank and a genius is that several years have gone by since the crank has made their unorthodox claim and no one believes them whereas a genius is someone whom the establishment believes after many years. Let's remember that Copernicus was considered a crank probably right up until Isaac Newton. Even in John Donne's poems (circa 1610) the geocentric universe is assumed.

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July 8th, 2014, 10:04 PM   #4
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Wasn't that done a hundred years ago?
Many believe that the attempt was not successful. Peter Smith for example calls the Principia "sloppy". Besides, it's quite easy to hoodwink a reader into believing that you've proven something, it's quite another to get a computer to output the results you want.
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July 8th, 2014, 10:09 PM   #5
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Actually, on second thought, since you've insulted me and called me a crank I guess I'll have to defend myself. So go ahead, let's hear why you believe Godels' theorems. Do you really understand what he was trying to prove or are you just parrotting mathematical dogma?
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July 9th, 2014, 05:18 AM   #6
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Actually, on second thought, since you've insulted me and called me a crank I guess I'll have to defend myself. So go ahead, let's hear why you believe Godels' theorems. Do you really understand what he was trying to prove or are you just parrotting mathematical dogma?
Goedel's incompleteness theorem is very simple -- some of the modern proofs (see, e.g., Aaronson for exposition) are quite short and easy to follow.

His second incompleteness theorem is essentially just a corollary to the first, trivial to prove given that.

Goedel's completeness theorem, I will admit, I have not proven. As I understand the proof is straightforward but rather involved.

Which of these do you doubt?
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July 9th, 2014, 05:47 AM   #7
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Goedel's completeness theorem, I will admit, I have not proven.
Good. So we've established that you called me a crank based on acceptance of dogma, not because you really understand that I'm a crank. That's the only thing that matters right now.
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July 9th, 2014, 11:14 AM   #8
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Good. So we've established that you called me a crank based on acceptance of dogma
Contingent on the assumption that you were speaking only of the completeness theorem, and that my partial understanding of the proof of that theorem being worthless.

I wouldn't say either assumption is justified.
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July 9th, 2014, 11:16 AM   #9
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Many believe that the attempt was not successful. Peter Smith for example calls the Principia "sloppy".
I would not call the Principia sloppy, but I was actually referring to the following work. Russell did a number on the Principia...

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Besides, it's quite easy to hoodwink a reader into believing that you've proven something, it's quite another to get a computer to output the results you want.
I've already mentioned several systems which have extensive computer-verified proofs. Metamath, in particular, has a quite weak internal framework which gives high confidence in its checking.
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July 9th, 2014, 11:19 AM   #10
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So in order for your ad hominem to make sense you must be rejecting (At least) Goedel's completeness theorem. So in particular you think there is a consistent countable first-order theory which has no model. Do you have an example, or are you taking this on faith?
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