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(I know you it doesn't let you, unless you have an alt account.) I must say that I am a little surprised that this is something you have not come across before

(I know you it doesn't let you, unless you have an alt account.) I must say that I am a little surprised that this is something you have not come across before and I take it you are surprised as well. Apparently foundations are not taught in every school. When we are talking about foundations we are usually concerned with something that prompted Euclid to write a very long time ago. In his Book 2 he defines geometric algebra for the first time but it is his parallel postulate which gets the most scrutiny because it is an outlier amongst his other claims. Elements is widely regarded as instrumental for logic and "natural philosophy" which is today called science. I'm the author of that paper construction I linked to. The thought experiment I am offering was something I invented 30 years ago and I have found much use for it in my studies of ontology. It is very simple and directly let me discover logics that do not rely on the law of the excluded middle and a rudimentary quantum logic. However its greatest strength is how it supports your intuition and gives legitimacy to your imagination. Cauchy sequences are what shows us that we have a way to actually reach every Real number, with no gaps, and this is what lets us talk about the number line as a mathematical object. Without this we don't have a foundation for differentiation. Perhaps now you can see why we might want to construct alternative number-like objects?

gjm11 2y ago
If your thought experiment involving pieces of paper has helped you to think of useful or interesting things that you wouldn't otherwise have thought of, that's great. It hasn't so far done anything of the kind for me. The fault could of course be mine. What I said was not (as you claimed in another comment in this thread) that I don't see how constructivism is relevant -- though in fact I don't think constructivism is relevant -- but that I don't see how your thought experiment about pieces of paper is relevant. I regret that it wasn't immediately apparent to me that your thought experiment was about constructivism. Now that you say it is, I can kinda see how it kinda relates, but for me thinking about your pieces of paper doesn't conjure up any useful insights about constructivism, and if I didn't already know about constructivism I don't think it would particularly point me in that direction. Again, if it works that way for you, good for you, but I think that when you present it to other people you are forgetting that the others (who unlike you haven't been thinking about this picture for 30 years) don't have the same associations with your imaginary pieces of paper that you have. I have no problem with the idea that one might want to construct alternative number-like objects. (And yes, I know what Cauchy sequences are and what one does with them. And no, they really don't have very much to do with the parallel postulate. There's an analogy between the parallel postulate and, say, the LEM, but that's all.) But (1) again, this has nothing much to do with Eric Lengyel's critique of how geometric algebra is presented -- when Lengyel talks about "foundations" he isn't talking about going down to the level of set theory, logic, HOTT, or whatever, he's talking about getting the basics of geometric algebra right when one already has notions like "real numbers", "vector space", etc., in hand -- and (2) your thought experiment happens not to do anything for me to clarify, inspire, etc., ideas about alternative number-like objects. (Also, it seems to me that if you want to use that thought experiment to say something about number lines then you want your 0-width lines, half-planes, etc., to lie across the number line, not along it.) It is simply not true that book 2 of Euclid "defines geometric algebra for the first time" or indeed defines it at all. Euclid is doing geometry but is not doing geometric algebra, which is something more specific.
- ganzuul 2y ago
  "What can we do with the black-on-white paper construction to extract information from the system?" I will need to leave this discussion with you here. I only mean to hold open the door for you and to give you two more variations of the tools that were used to define this long tradition of rational inquiry.

gjm11 2y ago

If your thought experiment involving pieces of paper has helped you to think of useful or interesting things that you wouldn't otherwise have thought of, that's great. It hasn't so far done anything of the kind for me. The fault could of course be mine. What I said was not (as you claimed in another comment in this thread) that I don't see how constructivism is relevant -- though in fact I don't think constructivism is relevant -- but that I don't see how your thought experiment about pieces of paper is relevant. I regret that it wasn't immediately apparent to me that your thought experiment was about constructivism. Now that you say it is, I can kinda see how it kinda relates, but for me thinking about your pieces of paper doesn't conjure up any useful insights about constructivism, and if I didn't already know about constructivism I don't think it would particularly point me in that direction. Again, if it works that way for you, good for you, but I think that when you present it to other people you are forgetting that the others (who unlike you haven't been thinking about this picture for 30 years) don't have the same associations with your imaginary pieces of paper that you have. I have no problem with the idea that one might want to construct alternative number-like objects. (And yes, I know what Cauchy sequences are and what one does with them. And no, they really don't have very much to do with the parallel postulate. There's an analogy between the parallel postulate and, say, the LEM, but that's all.) But (1) again, this has nothing much to do with Eric Lengyel's critique of how geometric algebra is presented -- when Lengyel talks about "foundations" he isn't talking about going down to the level of set theory, logic, HOTT, or whatever, he's talking about getting the basics of geometric algebra right when one already has notions like "real numbers", "vector space", etc., in hand -- and (2) your thought experiment happens not to do anything for me to clarify, inspire, etc., ideas about alternative number-like objects. (Also, it seems to me that if you want to use that thought experiment to say something about number lines then you want your 0-width lines, half-planes, etc., to lie across the number line, not along it.) It is simply not true that book 2 of Euclid "defines geometric algebra for the first time" or indeed defines it at all. Euclid is doing geometry but is not doing geometric algebra, which is something more specific.

ganzuul 2y ago
"What can we do with the black-on-white paper construction to extract information from the system?" I will need to leave this discussion with you here. I only mean to hold open the door for you and to give you two more variations of the tools that were used to define this long tradition of rational inquiry.