<html><head><meta http-equiv="Content-Type" content="text/html charset=utf-8"></head><body style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><div class="">Dear Carlos,</div>When I write P = {n | n is a prime number} and assert the existence of this set, I am asserting the existence of the CONCEPT that the set expresses.<div class="">I am not asserting the existence in some realm of all of its members. This is my way to handle the epistemology of set theory.</div><div class="">Set theory is a way to handle concepts. The concept of a prime number is not problematical in relation to any questions of existence.</div><div class="">The same is true of the concept 2^{N} = { S | S is a subset of the natural numbers} and all the other sets that can be expressed in ZFC.</div><div class=""><br class=""></div><div class="">With this understanding it is fun to play with the fantasy that the primes or the real numbers are “actually there” somewhere. This is fun and helps in thinking about them.</div><div class="">When I prove that a specific knot is knotted it excites me to realize that I have proved something about the infinity of instantiations of this knot, only a finite number of which I shall ever see. They all exist in potentia. </div><div class=""><br class=""></div><div class="">These rhetorical flourishes about potentia, possibles and adjacent possibles are a similar incitement to fantasy as we get a feeling about worlds unseen.</div><div class="">It is fun, but lets remember that just working with concepts and percepts is the main line.</div><div class=""><br class=""></div><div class="">Some concepts, maybe most, are tied up with their realizations. So if I am looking at Chess, I imagine the existence of all possible chess games. </div><div class="">"All chess games” is a concept just like “all primes”. That it happens to be a finite set does not make it more accessible than the infinity of the primes.</div><div class="">I can also indulge in the fantasy that I can “know” about “all chess games” but the only way to know about it is through a big sliice of experience with actual games and with </div><div class="">the hypotheses that people have made about general principles for the game.</div><div class=""><br class=""></div><div class="">The same is true for mathematics. The nice thing about infinite sets is that one has an open field for studying examples and finding principles that govern these examples.</div><div class="">Concepts are not “just” abstractions. You need to find out how they work in actual instantiations. I think the reason mathematics works even though people could argue about potential versus actual infinity forever, is that mathematics does not need to worry about existence of anything except concepts and to hope that the arena of concepts that it uses is consistent for the logic that is adopted for dealing with them.</div><div class=""><br class=""></div><div class="">And again I repeat, concepts come along with percepts. Every percept you have is accompanied by concepts and by the possibility of new concepts that can inform it.</div><div class="">Best,</div><div class="">Lou</div><div class=""><br class=""></div><div class=""><br class=""><div><blockquote type="cite" class=""><div class="">On Feb 28, 2024, at 10:43 AM, Carlos Gershenson <<a href="mailto:cgershen@gmail.com" class="">cgershen@gmail.com</a>> wrote:</div><br class="Apple-interchange-newline"><div class=""><meta http-equiv="content-type" content="text/html; charset=utf-8" class=""><div style="overflow-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;" class=""><div class="">About latent/Platonic information:</div><div class="">I think we can agree that there are (infinite) primes that have never been written down. Do they exist? I think so, in a particular way. Perhaps, as it was mentioned, the difference can lie between information that has been observed, and that which has not. Like the tree falling in a forest with nobody there to listen (observe), I guess we can assume that ontologically the tree fell, even if epistemologically no observer can say anything about that. (I elaborate on this distinction between “absolute being” and “relative being” here <a href="https://urldefense.com/v3/__https://arxiv.org/abs/nlin/0109001__;!!D9dNQwwGXtA!UTF2fg6sqvtQgQb_nuxoj5voh6N_Gw_p9V5p01SPx3bPOcoH28OuL7JzSKI1rsIztIhpoxXFZaIVGWmRwZo$" class="">https://arxiv.org/abs/nlin/0109001</a> although this is from my undergrad years and for sure could be updated).</div><div class=""><br class=""></div><div class="">Gordana: Isn’t ChatGPT an observer? I mean, LLMs can be said to be trained through “observation".</div><div class=""><br class=""></div>Best wishes,<br class=""><div class="">
<div style="letter-spacing: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><div style="letter-spacing: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><span class="Apple-style-span" style="border-collapse: separate; font-variant-ligatures: normal; font-variant-position: normal; font-variant-numeric: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-weight: normal; line-height: normal; border-spacing: 0px; -webkit-text-decorations-in-effect: none;"><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class="">Carlos</div></span></div></div>
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