Does mathematics actually model a continuum? I don't think so. If it did, it wouldn't lead to so many paradoxes and incorrect descriptions of experiences. Mathematics, I believe, provides rough models of discrete, measurable actions, which in themselves are practical for certain applications, but are also quite distant from experiences. — Rich
As far as I can tell, mathematics is totally reliant on the discrete, and because of this limitation constantly makes philosophical/ontological errors. Admission of this major limitation would allow philosophy to move ahead. As long as philosophers are pinned to mathematics, paradoxes will continue to confound. — Rich
So concretely, a discrete approach cannot uncover the nature of a continuous ontological reality. Other approaches must be used, and unfortunately current mathematics is simply not equipped. It is only adequate for discrete approximate measurements and predictions of non-living matter. It cannot be used to understand the nature of a continuous universe. — Rich
As I asked in the OP, is it possible to determine whether there are any real continua vs. everything (including space and time) being discrete? If so, how should we go about it? — aletheist
There is no a priori way of determining anything about reality ... — tom
Your insinuation that the real numbers cannot somehow model the physical continuum is rather odd. — tom
Of course mathematics can and does model a continuum. However, the accuracy and usefulness of such a model depend entirely on its purpose, and that is what guides the modeler's judgments about which parts and relations within the actual situation are significant enough to include. — aletheist
Is there any way that mathematics could evolve going forward that would enable it to deal with continuity more successfully? — aletheist
When you divide a line at a point, the point stays with one segment and not the other. As someone trained in math, it's hard for me to understand how this answer isn't satisfactory — fishfry
Can you explain (so that a philosophical simpleton like me could understand it) how mathematics has failed to successfully deal with continuity? — fishfry
When you divide a line at a point, the point stays with one segment and not the other. — fishfry
If you can divide the point on one of its sides ... — apokrisis
You can't subdivide a point and a point has no sides. It's sophistry to claim otherwise. If we can all agree on anything, it's that a point has no sides. — fishfry
So you defined a point as a howling inconsistency - the very thing that can't exist? The zero dimensionality that somehow still occupies a place within a continuity of dimensionality? — apokrisis
If we view the real numbers as specifying locations on a line, and we stop talking about points, perhaps things are less muddled. — fishfry
Perhaps, but it seems to me that we have then already conceded that the real numbers do not and cannot constitute a true continuum. They are now just labels that we have assigned to particular locations along the line, not parts of the line itself. — aletheist
When we mark a point on a line, we introduce a discontinuity. — aletheist
So this is why the Toms and Fishfrys are so content with what they learn in class. — apokrisis
Perhaps there's a sort of Heisenberg uncertainty between truth and precision. What we can say truthfully is imprecise; and what we can say precisely isn't true. — fishfry
I'll forego the opportunity to start a pissing match here and you earned a lot of good will with me for pointing me to the Zalamea paper. But really, was this necessary? Maybe I should just say fuck you or something. Would that serve any purpose? — fishfry
The answer is in the concept of "occupies a place." If we view the real numbers as specifying locations on a line, and we stop talking about points, perhaps things are less muddled. I agree with you that nobody knows how a line of dimension 1 can be made up of points of dimension 0. But math has formalisms to work around this problem. Would you at least agree that if math hasn't answered this objection, it's been highly successful in devising formalisms that finesse or bypass the problem? — fishfry
I'm used to a robust level of discussion in academic debate. — apokrisis
But this is an online forum, not an academic debate. Oops, putting it that way is likely to have the opposite effect of what I intend ... :s — aletheist
Look, you and fishfry are two of my favorite participants here, so I hope that we can all dispense with any unpleasantness and get on with what has already been a fascinating and helpful thread, at least from where I sit. — aletheist
It is like how you (as far as I can tell) take a deeply theist reading of Peirce, and I the exact opposite. But at the end of the day, the philosophy itself seems strong enough to transcend either grounding. So I can argue even angrily against any hint of theism while also conceding that it is still "reasonable" in a certain light. — apokrisis
Sure didn't read like that to me. Seemed a pretty brutal put-down, actually. — Wayfarer
So this is why the Toms and Fishfrys are so content with what they learn in class. To the degree that philosophy can still make a feeble groaning complaint about incompleteness, they feel utterly justified not to care.
Telling someone to fuck off is simply vulgar. — Wayfarer
this is why the Toms and Fishfrys are so content with what they learn in class. To the degree that philosophy can still make a feeble groaning complaint about incompleteness, they feel utterly justified not to care. They are trained within a social institution that had a purpose (hey guys, lets build machines!) and the very fact of having a definite purpose is (even for pragmatists) where an equally sharp state of indifference for what lies beyond the purpose to be fully justified.
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