Joe Niederberger posted Jun 29, 2013 2:04 AM > > Can you say what those propositions mean in any > precise sense? > > 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". > Specifically, which propositions?
>> i) Logic (in a sense) "IS INCLUDED IN" math; and
>> ii) Math (in a sense) "IS INCLUDED IN" logic.
>> a) Logic (in a sense) "INCLUDES" math; and >> b) Math (in a sense) "INCLUDES" logic. - -- or something else?
To be more precise about it, I would need to use what I call 'prose + structural graphics' (p+sg), which should become possible when I get my website up and running. Essentially, the structural graphics part of p+sg would enable us to 'integrate' in a fairly meaningful way your ideas on any issue of interest with mine (by working with structural models of our respective ideas), and this we claim would enable each of us to arrive at a better understanding of the other's ideas - AND (!!!) thereby of our own ideas (in the context of the other's ideas). That's the whole ball-game.
This is not possible to do in pure prose, to which we are restricted here - and to which we appear to be addicted. (Much as we once upon a time probably addicted to 'Roman Numeral Arithmetic' till we came to understand a hugely more simple process via 'placeholder arithmetic' + zero. Simple and 'obvious' concepts, but I believe they took a couple of centuries to 'get across' to people).
At least, this is my position: others no doubt may claim to be able to do it all in 'pure prose', but I frankly would generally not have the energy or the ability to follow any or all of such complex arguments.
I had quite some time ago read, with some care, the Stanford Encyclopedia of Philosophy (SEP) entry on "Principia" - and I have just now glanced at it again. (In fact, I had at one stage when I was much younger and mentally much more spry, struggled through sizable portions of "Principia" itself). The difficulties with the 'logicist program' of Whitehead and Russell (and other 'formalists') are indeed, in the main, to do with the axioms of "infinity" and "reducibility".
Accepted, but so what? If to read and understand the 'logicist position' is superhumanly difficult, then how do we mere mortals understand it - and, further, if we do understand it, how do we actually work with it on real-life issues? (Issues such as "To improve my results in my math exams"; "To reduce child malnutrition significantly in India"; and so on and so forth: THOSE are my only interests).
Questions about "reducibility" are generally found not to pose much difficulty when you look at issues via 'systems' (including 'thought-systems') using the Warfield approach. Questions about "infinity" are often to be found to be "infinite" in nature - and we may not (I believe but cannot prove) readily arrive at the end of them. I personally would just like to accept "infinity" as a highly usable 'concept', like "zero", and then to go about actually using it - leaving scholastics to the scholars.
In any case, I believe most such difficulties with the 'Logicist program' are 'reasonably' well resolved using 'Intuitionistic Logic' (as may generally be gauged from most of what constitutes 'my approach', which develops from 'Warfield graphics' when applied to 'systems'). In general, I like to look at issues to start with at least using the simplest of common sense, and then apply some 'systems thinking' to them. This approach, I have found, helps resolve a fair number of very knotty issues (not all, I accept). OK, I don't demand that ALL issues be resolved - all I want to do is to go ahead and get a few things done, in practice on the ground.
[I am NOT here claiming that the 'Intuitionist approach' is the 'end of all things', or that all our answers will come out of that. We have a LONG way to go before we can claim any such powers - and we (humans) may well destroy the possibility of our survival on this or any other planet before we reach that stage: i.e., such issues may quite swiftly become somewhat moot, unless we discover ways to continue life on earth].
I know I've not adequately answered your request - but I hope I have at least indicated how we might go about answering it (if we had the facilities to search out needed answers).