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You did, thank you for the discussion! I actually have a first edition of Keisler's "Elementary Calculus," well-worn I assure you; throughout my mathematics deg

You did, thank you for the discussion! I actually have a first edition of Keisler's "Elementary Calculus," well-worn I assure you; throughout my mathematics degree, I sought various other methods of approaching the curricula. I'm very much a fan of Robinson's program of infinitesimals; occasionally, I'm treated to even more rarefied notions of these beloved ghosts of departed quantities. https://home.sandiego.edu/~shulman/papers/sdg-pizza-seminar.pdf https://home.sandiego.edu/~shulman/papers/sdg-pizza-seminar.... Here's one I find particularly fun, wherein you disregard law of excluded middle to use nilpotents! I'm currently trying to digest "Geometric Algebra: An Object-Oriented Approach." It is a real pleasure to see you mention some of my pet favorites; thank you, thank you, a hundred times over friend! You've put a song in my heart today. EDIT: I was so wrapped up in having met a fellow traveler that I forgot to leave an "in," should you wish to continue this thread. I also find it a tragedy that my foray into the constructivist and sundry corners of math was relegated to self-study. Having approached a few professors on this topic, the general consensus seems to be "why waste your time?" To what do you think the intuitive power of these equivalent-but-alternative foundations is owed? Personally, I think it has something of the basis that colors the division of the analytically- vs algebraically-minded; yet even the cause of this is a mystery! (However, let me be careful here: I don't wish to give the impression of undue competence. I'm broadly-read, but woefully underskilled.)

goldenkey 5y ago
Thank you for sharing SDG. I haven't heard it mentioned in many years. The simplification power it provides is beautiful! [1] I'm elated over having delighted and filled you with music. Your kind words have made me beam from ear to ear as well :-) I think the intuitive power of these other formulations comes from the spatial/geometric imagery that these disciplines naturally provide. Another aspect is the conversion of unwieldy processes into simpler objects. Like nonstandard analysis replacing the limit process with the infinitesimal object. The paradigm of replacing large processes with things that can be intuited might not be very profound if as programmers we bring up first class functions. A function is just an object anyhow! True but the human mind seems to be less efficient at composing functions than composing objects. So though it's all interchangable, we seem to work better when we are given mentally ergonomic foundations. Back to what you said in your previous posts, the ease of manipulation in these alternative theories, it probably lends a bit from this exchange of complicated processes for simpler things. [1] https://en.wikipedia.org/wiki/Synthetic_differential_geometry https://en.wikipedia.org/wiki/Synthetic_differential_geometr...

goldenkey 5y ago

Thank you for sharing SDG. I haven't heard it mentioned in many years. The simplification power it provides is beautiful! [1] I'm elated over having delighted and filled you with music. Your kind words have made me beam from ear to ear as well :-) I think the intuitive power of these other formulations comes from the spatial/geometric imagery that these disciplines naturally provide. Another aspect is the conversion of unwieldy processes into simpler objects. Like nonstandard analysis replacing the limit process with the infinitesimal object. The paradigm of replacing large processes with things that can be intuited might not be very profound if as programmers we bring up first class functions. A function is just an object anyhow! True but the human mind seems to be less efficient at composing functions than composing objects. So though it's all interchangable, we seem to work better when we are given mentally ergonomic foundations. Back to what you said in your previous posts, the ease of manipulation in these alternative theories, it probably lends a bit from this exchange of complicated processes for simpler things. [1] https://en.wikipedia.org/wiki/Synthetic_differential_geometry https://en.wikipedia.org/wiki/Synthetic_differential_geometr...