No, you claimed the reals can be disordered and made discrete. — tom
The reals in their usual order are a continuum. They can be reordered to be discrete. — fishfry
If by continuum we mean a particular philosophical idea of a continuous space, then the mathematical real numbers may or may not satisfy a philosopher. If by continuum we mean the standard mathematical real numbers, then we are being circular. Certainly the standard real numbers are not a proper model of the intuitionistic continuum. — fishfry
As far as I can tell, mathematics is totally reliant on the discrete and because of this limitation constantly makes philosophical ontological errors. — Rich
I am not convinced that this is true. Two of Peirce's major objectives for philosophy were to make it more mathematical (by which he meant diagrammatic) and to "insist upon the idea of continuity as of prime importance." Surely he must have considered these efforts to be complementary, rather than contradictory. — aletheist
I am not convinced that this is true. Two of Peirce's major objectives for philosophy were to make it more mathematical (by which he meant diagrammatic) and to "insist upon the idea of continuity as of prime importance." Surely he must have considered these efforts to be complementary, rather than contradictory. — aletheist
So concretely, a discrete approach cannot uncover the nature of a continuous ontological reality. — Rich
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
... he attacked the Cantor-Dedekind philosophy of the continuum on the ground that it was committed to the reduction of the continuous to the discrete, a program whose philosophical cogency, and even logical consistency, had been challenged many times over the centuries.
So regular maths is "wrong" in always framing reality in constructivist terms. And yet in the end maths is a tool for modelling. We actually have to be able to calculate something with it. And calculation is inherently a constructive activity. — apokrisis
So while we can sketch a picture of systems of constraints - like Peirce's diagrammatical reasoning - that is too cumbersome to turn into an everyday kind of tool that can be used by any schoolkid or universal turing machine to mechanically grind out results. — apokrisis
one should never take too much notice of a mathematician making extrapolations of a metaphysical nature. They are bound to be misguided just because they hold in their hands a very impreessive hammer and so are looking about for some new annoying nail to bang flat. — apokrisis
So I would still like to see the alleged proof that the real numbers form a true continuum as Peirce defined it, which (as I understand it) is similar but not identical to the intuitionistic continuum. — aletheist
Please identify a natural number or integer that is not capable of being counted. — aletheist
Instead, what I am arguing is that it is possible in principle to count all of the natural numbers (and integers) because it is possible in principle to count up to and beyond any particular natural number (or integer). — aletheist
As far as I can tell, mathematics is totally reliant on the discrete and because of this limitation constantly makes philosophical ontological errors. — Rich
Can you (or anyone) supply some of relevant Bergson and Pierce links that would shed light on the relation between the mathematical real numbers and the philosophical idea of the continuum? — fishfry
Can you supply some of relevant Bergson and Pierce links that would shed light on the relation between the mathematical real numbers and the philosophical idea of the continuum? — fishfry
But first, discrete must be discarded in the realm of ontology. — Rich
Yes, there is such thing as continuity and we experience it quite concretely as duration (real time). — Rich
The highest number is the one that's not capable of being counted. — Metaphysician Undercover
And which number would that be? I asked you to identify it, not describe it. — aletheist
The accepted mathematical one from set theory, "able to be put into bijection [one-to-one correspondence] with the natural numbers."
My notion of "potentially countable" or "countable in principle," which is that there is no particular largest value beyond which it is logically impossible to count.
The notion of "actually countable," which requires it to be possible to finish counting.
You have made it quite clear by now that you reject the first two, but that does not render them false or contradictory - just different from yours. — aletheist
A dream, therefore is as real as anything else. — Rich
'Real' is a word invented in the 13th century to signify having Properties, i.e. characters sufficing to identify their subject, and possessing these whether they be anywise attributed to it by any single man or group of men, or not. Thus, the substance of a dream is not Real, since it was such as it was, merely in that a dreamer so dreamed it; but the fact of the dream is Real, if it was dreamed; since if so, its date, the name of the dreamer, etc. make up a set of circumstances sufficient to distinguish it from all other events; and these belong to it, i.e. would be true if predicated of it, whether A, B, or C Actually ascertains them or not. The 'Actual' is that which is met with in the past, present, or future.
It would be false to say that something which is not capable of being counted is countable. — Metaphysician Undercover
Therefore we can conclude that the set of natural numbers is not countable. — Metaphysician Undercover
And yet set theory explicitly says otherwise. — aletheist
Therefore we can conclude that the set of natural numbers is not countable.
β Metaphysician Undercover
And yet set theory explicitly says otherwise. — aletheist
It appears like your set theory, if it really is as you describe, relies on the fallacy of composition. — Metaphysician Undercover
If counting is an activity that takes place in time, then a finite universe doesn't give you enough time to count any more than some finite number. There are 10^80 hydrogen atoms in the universe. That's a very small natural number. You can't count it. — fishfry
I've got a lot of reading to do. I'm afraid I can't pick up those many volumes that have been suggested, but I will definitely Google around. — fishfry
Please locate that quote of mine, I can't find it and don't remember saying it. — fishfry
The reals in their usual order are a continuum. They can be reordered to be discrete. — fishfry
Look, we have been using at least three different definitions of "countable" in this thread:
The accepted mathematical one from set theory, "able to be put into bijection [one-to-one correspondence] with the natural numbers."
My notion of "potentially countable" or "countable in principle," which is that there is no particular largest value beyond which it is logically impossible to count.
The notion of "actually countable," which requires it to be possible to finish counting. — aletheist
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