It is simulation... [but] mimesis is not science, it only mimics science." — StreetlightX
But nevertheless the emphasis on pure reason, on things that could be known simply by virtue of the rational mind, is one of the major sources of Western philosophy. — Wayfarer
I think trying to attribute all of that to Pythagoras [if that is what is being said] is at the very least drawing a long bow. — Wayfarer
But nevertheless their emphasis on pure reason, on things that could be known simply by virtue of the rational mind, is one of the major sources of Western philosophy. I haven’t read the paper and am not inclined to, but calling it ‘a disaster’ seems to me nothing more than empiricist polemics. So, sure, the particular notion in question might be completely wrong-headed but write off Pythagoreanism and you’d be having to communicate your OP via ink or smoke signals. — Wayfarer
"It is my contention that mathematics took a disastrous wrong turn some time in the sixth century B.C" — StreetlightX
Seem to me, if the result of that disaster is mathematics, science and technology, we could use more such disasters. — Banno
The conclusion to be drawn is not that mathematics is useless or somesuch; only that one should engage in mathematics without the (unscientific) assumption of commensuribility. Or put otherwise: don't assume the commensuribility of everything in the world (this is theology); instead, run tests, inject a good dose of empiricism and pay close attention to whatever phenomenon you aim to examine. — StreetlightX
Non linear maths demonstrated that measurement error is not necessarily linear. — apokrisis
AFAIK the number line is complete (wholes, rationals, integers, rationals, and irrationals). We even have the real-imaginary number space. — TheMadFool
The whole problem is precisely over the question of 'completion'. The assumption of commensuribility turns on the idea that, once we 'complete math', we could then use mathematical tools to create a one-to-one model of all reality (Rosen: "The idea behind seeking such a formalized universe is that, if it is big enough, then everything originally outside will, in some sense, have an exact image inside"). But if commensuribility does not hold - if not everything in the universe is in principle able to be subject to a single measure - then no such 'largest model' can exist. — StreetlightX
Importantly this does not mean that modelling is a lost cause; instead, it means that modelling must be specific to the phenomenon so modelled: beyond certain bounds and threshold values, modelling simply ends up producing artifacts (at the limit, you get Godel's paradoxes!). You can have models of this and models of that but never THE model. — StreetlightX
Does current science have commensuribility as a principle? — TheMadFool
Given the existence of irrationals, isn't the point made here already accepted? The existence of irrationals has been known since ancient times, as you say.
How does the Pythagorean doctrine of commensurability lead to Zeno's paradoxes? — Snakes Alive
Doesn't the principle just fall out as a corollary of the infinite divisibility of length (or any other measurable relation)? — gurugeorge
AFAIK the number line is complete (wholes, rationals, integers, rationals, and irrationals). We even have the real-imaginary number space. — TheMadFool
So what the Zeno paradoxes essentially mark is the irreducibly of incommensuribility. Making the irrationals the limit of a converging series of rationals in order to save commensuribility is a bit like trying to suppress a half inflated balloon: short of breaking the balloon, all one can ever do is shift the air around. One of the take-aways from this is that the very idea of the (continuous) number-line is a kind of fiction, an attempt to glue together geometry and arithmetic in a way that isn't actually possible (every attempt to 'glue' them together produces artifices or problems, either in the form of irrationals, or later, in the form of Zeno's paradoxes - and, even further down the line, Godel's paradox). — StreetlightX
Just wanna come back and address these together as they all hit on similar points that I think deserve to be expanded upon. The idea as I understand it is this - there is in fact one way to 'save' the assumption of commensurability after the introduction of the irrationals, and it is this: to treat irrationals as the limit of a convergent series of rational numbers. In this way, we don't actually have to deal with incommensurate values per se, only rationals (Rosen: "At each finite step, only rationalities would be involved, and only at the end, in the limit, would we actually meet our new irrational. This was the method of exhaustion...") — StreetlightX
Just wanna come back and address these together as they all hit on similar points that I think deserve to be expanded upon. — StreetlightX
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