GS Chandy
Posts:
7,941
From:
Hyderabad, Mumbai/Bangalore, India
Registered:
9/29/05


Re: Is logic part of mathematics  or is mathematics part of logic?
Posted:
Jun 30, 2013 2:04 PM


Joe Niederberger posted Jun 30, 2013 1:29 AM (GSC's remarks interspersed): > Joe N says: > >Take a look at the Stanford article on "Principia". > The main problems with the logicist program were > perceived to be the "axiom of infinity", and that of > "reducibility". > > OK, let me try again, I was just trying to get you to > reiterate in your own words what your "propositions" > meant to you, while touching on the fact that there's > a big long history here. > As stated in my earlier post, to reiterate or clarify in "my own words" what those propositions mean to me would demand 'prose + structural graphics' (p+sg). In 'pure prose', any such iteration would be nearfutile and a real waste of time and effort.
For any complex argument, I claim that 'pure prose' is, in fact, futile if not accompanied by 'structural graphics'. I cannot explain THIS proposition unless you know the 'language' that develops from this tiny extension to 'prose' (i.e. p+sg). Though we are both now communicating in English, I claim that 'pure prose' is insufficient to communicate in any effective way the structural complexities of any argument put forth.
Let me try to explain by way of an analogy. Do you think anyone would be able to understand, say, calculus if all we had was 'arithmetic using Roman numerals'? No 'zero; no placeholder arithmetic. (I mean understand and then USE in reallife examples). Yes, I know that Archimedes had worked out 'The Method' long before such arithmetic become widely used. I understand 'The Method' is said to contain ALL the foundations of calculus  but that was Archimedes doing it , not GSC.
(By the way, I cannot see myself  and 99,999 out of a million people  being able to understand AND effectively use calculus in this way  I mean working out reallife problems, not just a vague understanding of the 'principles'. Yes, I would probably understand the principles  but I'm pretty certain I wouldn't be able DO anything PRACTICAL with it). No, I have not examined 'The Method'.
Just so, I claim that it is in fact extremely difficult  perhaps even impossible  for most of us to understand 'systems'  AND use it in practical life  without p+sg. Kenneth Boulding, Ludwig von Bertalanffy (and others) developed the concepts of 'general systems theory' (GST). However, it needed a special language for GST to become usable in real life, i.e., to become a usable science. (I understand that, when von Bertalanffy initially propounded the concepts of GST, some clown tried to satirise him by writing a book called "General Systemantics").
Now, explanation of those propositions with the "inclusion" relationship does require some p+sg. Doubtless we could go into unendingly long explanations of them in 'pure prose'  but it would be just a lot of foolishly wasted effort: for me to try to write it all up; for you to read up what I've written. > > Here's the deal as I understand it, very very > roughly, in my own words. Don't be offended by my > lecturing style, > or hold it against me if you already know all this > stuff, I'm just laying it out in case anyone wanted > to challenge these understandings or delve deeper > into something more specific or somewhat different > than your propositions (about "One IS INLCUDED IN the > other.") > The relationship "IS INCLUDED IN" (/INCLUDES) is transitive. Discussion of ANY transitive relationship is effective and 'quite efficient' using p+sg. It is highly ineffective (and inefficient) using pure prose. In order to help 'delve deeper', p+sg recommended. Huge waste of thinking effort in trying to do this via 'pure prose'. > > Logic is that which seeks to understand the "laws of > thought", independent of subject matter. So if > we agree that the following is logical: > > 1. All men are mortal, and > 2. Socrates is a man, then > we must agree with Woody Allen that > 3. Therefore, all men are Socrates. > > If I replace "All men are Mortal" with > 1. All toves are slithy... > > The logic of the syllogism is the same. > OK. All this is well known. > > Obviously also, the above is not generally considered > math. So its not generally considered reasonable to > say that "Math includes all of logic" or "All logic > is just math." > > But the reverse proposition, that "all math is just > logic" is less obviously false, and was seriously > considered at the end of the 19th century. > Accepted more or less in full (with a couple of reservations, mainly relating to the fact that such ideas are NOT at all effectively  or efficiently  discussed in 'pure prose'). For instance, it is difficult to discuss "INCLUSION" effectively in 'pure prose'. (You already know I'm sure about 'the enhancement of understanding' of the concept of "INCLUSION" that develops via Venn diagrams. I claim that SIGNIFICANTLY more understanding develops when we use p+sg. > > Now, I'll cut to the chase, and be somewhat purposely > vague. It seems to me, the big stumbling block, is > the subject matter of mathematics. The axiom of > infinity, for example, claims "there is an infinite > set". This notion of "actual" or "completed" > infinities has along history going back to the days > of Aristotle. > Having once spent practically every waking (and sleeping) hour thinking about math, I'm reasonably aware of its history as well. > > Gauss seems to have railed against the > concept, saying actual infinities are *never* treated > in mathematics, yet a hundred years later most > everbody is accepting them. Strange! (You mention the > intuitionists  they disagreed with the changing > fashion and are still a minority.) > Not so strange at all. I mean the phenomenon that 'actual infinities' are nowadays 'justifiably' accepted [and even used, though, in the main, intuitively] by a great many people where Gauss once railed against them.
This is, to my mind, not strange at all  but something that could probably have been predicted by a person with adequate insight even in the 1800s [when Gauss railed against them].
Fashions change, as in this very instance  and as seen by the phenomenon you commented on above. > > If you dig into the "reducibility" axiom, its a lot > harder to understand, but I think it comes down to a > similar thing  just what is it the mathematicians > are talking about, really? What is a "set"? When do > we know we have something really mathematizable, > versus something plagued with paradox? What are the > rules to avoid paradox? > Accepted. Except that I don't know if there can be rules to avoid paradox. 'Set theory' had, I accept, helped clarify some paradoxes. But there are paradoxes beyond those clarified in set theory. For instance, the idea of the 'Axiom of choice' and its equivalent Zorn's Lemma also leads to some deeper paradoxes, I understand. (I'm not very uptodate on this, but I did see some mention of this somewhere). > > Anyway, for whatever its worth, and this is offhand, > I could easily be shown wrong, but I have a lot of > sympathy and respect for the logicist, intuitionist, > formalist, and computationlist movements. > As have I. I use any or all these schools, as I find it intuitively appropriate. However, despite all the profundities that these schools have explored, I am confident that "There are more things in heaven and earth, Horatio, than are dreamt of in your philosophy" (and in your science, and your math, and your....). > >On the > other hand, I have a picture of Kurt Gödel hanging by > my desk, and he was a famous Platonist. Still he was > a logician, and I think the *best* part of > mathematics is *logic*, actual infinities or no > actual infinities, sets or no sets. > I'd *broadly* accept your proposition above if you'd substitute the descriptor "fundamental" for *best*  but I would also insist that any prose discussion of these concepts is likely only to lead us into foncusion. We do need p+sg for any effective and efficient discussion  i.e., a discussion that is conducted in less than an 'infinite' period of time; AND that leads to adequate understanding:
a) by me of the complex arguments you may put forth; and b) by you of the complex arguments that I may put forth. > > Here's a cute paper: > http://www2.gsu.edu/~matgtc/three%20crises%20in%20math > ematics.pdf > I've downloaded the paper  shall read it in due course. > GSC >  End of Forwarded Message
 End of Forwarded Message
Message was edited by: GS Chandy
Message was edited by: GS Chandy

