aletheist

Citation, please. On the contrary, Peirce does not dismiss points/instants, he clarifies that they are creations of thought rather than real constitue...

There are no "wrong" assumptions in pure mathematics. It is the science of reasoning necessarily about hypothetical states of things. LEM holds for an...

I laughed out loud at this. It reminds me of my favorite line from the movie, The Incredibles 2, when Bob is trying to help Dash with his homework: "W...

Heh, I like it! I am an engineer, not a mathematician, so I would welcome your thoughts on SDG/SIA--although they probably belong in a new thread. Not...

No, I do not agree with this, and neither did Peirce. Infinitesimals/moments are indefinite, not distinct. The principle of excluded middle does not a...

I have come across some arguments that teaching calculus using SDG/SIA--grounded in category theory, rather than set theory--is more effective than ei...

There is nothing inherently contradictory about the mathematical concept of an infinitesimal, which is not necessarily defined as 1/?. Again, if you t...

Again, Peirce himself considered his realism to be "extreme." That does not seem very "nuanced" to me. :smile:

That is exactly what I described as discrete time, not what you described. Each individual frame of such a movie corresponds to an instantaneous state...

In other words, continuous lapses of time with finite duration, arranged such that each one starts when the previous one ends. Calling this "discrete ...

With all due respect, this sweeping generalization is rather misleading. Peirce described his own view as "extreme scholastic realism" (CP 8.208; c. 1...

On the contrary--if time were discrete, then it would necessarily consist of durationless instants at some fixed interval. The fact that "now" cannot ...

As long as you continue to insist on this, there is nothing more for us to discuss.

By that reasoning, an infinite set could have logical existence if our universe was infinite. But logical possibility is not at all dependent on actua...

Thanks for this. I would add that the same is true of the Planck time, since it is defined as the duration required for light to travel the Planck len...

Please provide citations or (better) quotes from Peirce's writings to substantiate your assertions about his views, as I did for mine. I remain unconv...

Peirce does not "replace" points with infinitesimals; they are two different concepts, and there is still a role for points--not as the parts of a lin...

I am largely employing the terminology and philosophy of Charles Sanders Peirce in all this, so I will offer a couple of his examples. If I were to ho...

Yet again: mathematical existence does not entail metaphysical actuality. Within mathematics, the number 10^100 (1 googol) indubitably exists and is a...

Okay, but you and I agree that this now-standard mathematical definition of a continuum is philosophically faulty; in my case, because I hold that a l...

In my view, nothing within mathematics is actual--again, it is the science that reasons necessarily about strictly hypothetical states of affairs--and...

Sure, but in the quoted text, he did not claim that there is an actual set containing all the natural numbers. And his (incorrect, in our view) belief...

No, it is defined as a potential infinity. One more time: mathematical existence does not entail metaphysical actuality. No one, except perhaps an ext...

Yes, given the standard mathematical definitions, the proposition that the number denoted by "5" possesses the character denoted by "prime" is true. D...

You simply refuse to acknowledge the definitions of terms that others are employing, and thus consistently (and persistently) attack straw men. Actual...

All of these are logically possible, just not metaphysically possible. No one is claiming otherwise. When mathematicians state that the natural number...

This indicates a confusion between existence in mathematics and actuality in metaphysics. They are not synonymous or equivalent. Everything that "exis...

False. Again, if space is a continuous whole, then it is not composed of individual and distinct positions.

The wrong assumption in this case is that the true continuity of space would require your hand to complete an actually infinite number of steps by pas...

No, the only actual intermediate positions are the ones that we individually mark. There is a potential infinity of such positions, but we can only ma...

No, it only marks positions 0 and 1; marking any individual intermediate positions would require their explicit designation, and we can only ever do t...

How much of Peirce's metaphysics (and mathematics, and phenomenology, and logic/semeiotic) have you actually studied carefully? What fundamental disti...

This is the same mistake that Zeno made. Positions 0 and 1 do not exist unless and until we arbitrarily mark them as such, and the same is true of any...

No, it would not. If space is truly continuous, then it is not composed of distinct positions. We arbitrarily impose distinct positions on space for v...

Only those who (perhaps naively) embrace Cantor's mathematical model of a continuum as isomorphic to the real numbers would affirm this metaphysical i...

Fair enough, thanks.

If you adamantly deny the reality of space and time, then there is nothing more for us to discuss on that front. The issue is not so much the mathemat...

This confuses an abstract idea with its object--i.e., what it represents. The fact that the concepts of space and time account for what we observe doe...

Peirce's rather poetic analogy between semeiosis and music in "How to Make Our Ideas Clear" (1878) seems relevant here: "In a piece of music there are...

This is a common misconception. What modern science has demonstrated is that there is a smallest observable unit of space (and time), which does not e...

Thanks for the correction.

Please define your terms and assumptions. What is "time"? What is a "moment"? Why do you think that "infinite time between any two moments" is impossi...

I was thinking of the generalized continuum hypothesis--the idea that for any infinite set of a given cardinal (aleph_n), its power set is always of t...

Or as Charles Sanders Peirce aptly put it, mathematics is the science that draws necessary conclusions about purely hypothetical states of things. Iro...

No, mere complexity is also insufficient; "intelligent design" theory requires specified complexity and/or irreducible complexity to count as evidence...

"Intelligent design" theory acknowledges that mere order is insufficient, instead requiring "specified complexity" to count as evidence of design. Its...

Charles Sanders Peirce denied the existence of God, but argued for the reality of God. Something exists if it reacts with the other like things in the...

Or, logic is simply consistent with the eternal and immutable nature of God, rather than something primary to God or created by God. This is basically...

Thanks for this. Just curious, is there a constructive argument for the lemma, such that it also holds for intuitionistic logic where proof by contrad...

What we tend to dislike about certain people is not their actual (finite and imperfect) love, power, or knowledge, but their ways of expressing their ...