Timeline for What is the status of irrational numbers within finitism/ultrafinitism?
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17 events
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| Jul 16, 2012 at 16:24 | comment | added | Mirco A. Mannucci | PS Your last comment, whether tongue-in-cheek or not, on my poor puny little brain versus my super-smart laptop, is also very good. You touched on the fundamental point: for a being who can compute till say 100, 1000 is de facto IN-FINITE. | |
| Jul 16, 2012 at 16:19 | comment | added | Mirco A. Mannucci | I thought your story was superlative. Really, really good. So, that seems to suggest that you understand well my crazy point of view. Then, why not leaving for a while Cantor's lofty paradise and joining the underground, just for a change? I can guarantee you one thing: that down there you will find a miniature version of everything in Plato/Cantor's attic, and in fact much more (even things that the ultra-infinitists do not dream about, like numbers bigger than the alephs). You will find, I trust, one surprising truth, this one: the finite is not so small, after all.... | |
| Jul 16, 2012 at 16:10 | comment | added | Mirco A. Mannucci | @Tim, again, very true. But you seem to forget that I am not an actualist, I have no problems with a free flight into the imaginary realms, as long as I know that they are imaginary. What I do have problems with is pretending that those imaginary flights (by the way, one of those extrapolates from large finite/concrete totalities to the mythological N) are items in Plato's attic. One final note for you dear Tim: I remember a few months ago, during the FOM blahblah on Nelson's alleged proof, you wrote a little story in which you play some crazy dude like me, who does not believe in numbers. | |
| Jul 16, 2012 at 13:50 | comment | added | Timothy Chow | @Mirco: You seem to be assuming that the laptop is actually running through a large number of computational steps. But the number of steps is ungraspably large for your meager human brain. Only via a fictional, imaginary process of extrapolating your ordinary experience to an ungraspably large realm are you justified in believing that the laptop is doing...something, who knows what? | |
| Jul 16, 2012 at 10:11 | comment | added | Mirco A. Mannucci | , intuitionistic logic is the logic of a relentless, tireless immortal mathematician with unbounded resources, but the soon-to-be ultrafinitistic logic is the logic of concrete, resources-bound machines (NOT Turing machines!), and concrete humans. | |
| Jul 16, 2012 at 10:06 | comment | added | Mirco A. Mannucci | Tim, absolutely correct on the premise: my lap is much much better than I am. Only I do not see your "so". I want to compute something: I write my program, I verify it (to the best of my very modest capabilities) and I let my lap run. Why should it be different? There is a much greater chance that I make a mistake during the computation of, say, the first 1000 digits of PI using the formula above, than writing a few lines matlab routine which inputs 1000 and spits them out. CODA for you Tim, and the other FOM fellows : Classical Logic is the logic of an omniscient God | |
| Jul 16, 2012 at 0:03 | comment | added | Timothy Chow | Mirco, your laptop is much better at computing than you are, so I don't see on what basis you believe everything it believes. It would seem that you should only believe the computations that you can verify yourself, without asking the computer. | |
| Jul 15, 2012 at 16:28 | comment | added | Mirco A. Mannucci | I think you are quite right, my laptop does not believe in anything, it just knows (when programmed rightly ) how to compute, and so do I. It does not need any ontology, neither do I (nothing wrong with ontology, of course, I am a big fan of Plato, just do not see any reason to call on him when talking/doing math) | |
| Jul 15, 2012 at 6:14 | comment | added | abo | I don't think laptops believe anything. If you claim they did, then it would seem they also "believed" in numbers. | |
| Jul 14, 2012 at 21:43 | comment | added | Mirco A. Mannucci | Abo, ask my laptop. Whatever it believes, I believe it too.... :) | |
| Jul 14, 2012 at 19:57 | comment | added | abo | Mirco, what formulas do you accept to exist (what formulas are there)? | |
| Jul 14, 2012 at 19:20 | comment | added | Mirco A. Mannucci | The point, Godelian, at least how I see it, is that COUNTING (or more generally, computing) is what matters. In the beginning was counting, not the natural numbers.... | |
| Jul 14, 2012 at 19:18 | comment | added | Mirco A. Mannucci | applies to PI, to Mahlo's cardinals, etc. The only difference between 5 and PI or Mahlo is simply the rules of the game. If on the other hand you mean that 5 is more "graspable" than, say, graham's number, the answer is still yes, in the sense that the corresponding "in progress" equivalence classes of provably equivalent terms are quite different. The class 5 has plenty of terms, and also I know how to compare it to other classes, such as "7", "10000", etc. The class "graham" less so. | |
| Jul 14, 2012 at 19:11 | comment | added | Mirco A. Mannucci | Godelian, if what you mean is that I know how to count up to say 5, and understand that a set of 5 pears is the equinumerous to a set of 5 apples, yes. If what you mean is that I understand what a wff formula is, the answer is still yes (in both cases, my laptop understand it too). If what you mean is that I understand what the "number" 5 is in Plato's attic, the answer is no. To me 5 is an eternal mystery, or to put it more mathematically, a constantly expanding equivalent class of terms which happen to be proved/computed as equivalent by the logic of the arithmetical game. Same of course | |
| Jul 14, 2012 at 18:39 | comment | added | godelian | Mirco, if one understand syntax and is therefore capable of distinguishing between different entities (like marks on a piece of papers), isn't one also capable of grasping the ontology of a (at least small) natural number? | |
| Jul 14, 2012 at 17:52 | history | edited | Mirco A. Mannucci | CC BY-SA 3.0 |
added 338 characters in body
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| Jul 14, 2012 at 17:39 | history | answered | Mirco A. Mannucci | CC BY-SA 3.0 |