But, it is right with present modern limit theory and technology applied in this proof that we meet a “strict mathematical proven” modern version of Ancient Zeno’s Achilles-Turtle Race Paradox: the “brackets-placing rule" decided by limit theory in this proof corresponds to Achilles in Zeno’s Paradox and the infinite items in Harmonic Series corresponds to those steps of the tortoise in the Paradox. One remarkable aspect of John Bell's contribution is that such an algebraic notion can be presented without resorting to the language of category theory at the same time that small book is plenty of philosophical remarks which do not ignore the history of both mathematics and philosophy.Įach operation in this proof is really unassailable within present science theory system. In parallel with the formal, axiomatic, presentation of infinitesimals, there is also a number of papers in contemporary philosophy of mathematics which already discuss the issue in precise terms. Such an algebraic notion makes no reference to transfinite sets. But the most radical step, and perhaps the philosophically most important one in recent times, was the purely algebraic notion which came with synthetic differential geometry, by means of category theory - this line of thought implies the rejection of the logical principle of the excluded middle. Jerome Keisler wrote a clear textbook of Calculus based on infinitesimals. A new start came with Abraham's Robinson's re-introduction of infinitesimals by means of logic (more specifically, through the compactness theorem as a tool in model theory to construct non-standard reals). Let me just remind Bertrand Russell's description, at the beginning of XXth century, of Zeno's paradoxes in terms of set theory: it is also a reference point in order to understand the links between mathematical and philosophical considerations about infinitesimals and this answers your last question, even though it does not mean that Russell's analysis must be subscribed. The literature on infinitesimals is huge. The history of the Calculus was also due to an effort towards conceptual clarification, which is something of interest for philosophy, whereas it now seems that the whole set of issues has to be restarted from scratch with much less effort. It just seems that the history of both mathematical and philosophical understanding of the continuum is totally neglected. So, is there a mathematically rigorous statement of a problem in this thread, or not?įirst, I agree with what Wes Raikowski already remarked above: the questions as they are raised here are too vague. If this thread is not about mathematics, and is about, say, philosophy, okay, but the risk is lack of precision arising from how different individuals may not use the same words to mean the same thing, and there is no authority to whom the philosophers might appeal, except other philosophers. Geng Ouyang, Sir: There exists an article by Solomon Feferman "Conceptions of the Continuum" that discusses at least a dozen conceptions, including "The continuum in infinitesimal analysis." My view is that there are numerous alternative (and very interesting) mental models of "the continuum," some of which explicitly include mathematical microlects with a notion of "infinitesimal." To me there is no useful question about which one is "right." Just as one cannot appreciate a sculpture without viewing it from enough angles and positions to have seen its entire surface, one cannot comprehend the continuum - including its infinitesimal parts - without some mastery of several alternative (preferably mathematical) microlects.
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