Earman and norton 19991/11/2024 ![]() In hypertasks, the number of operations is uncountably infinite (Clark & Reed, 1984 Al-Dhalimy & Geyer, 2016). But (P6) is the standard position, and we will run with it here.įor our next metamathematical trick, we make use of the idea of a supertask, that is, tasks, consisting of a countable infinite number of steps or operations, that are completed in a finite time (Manchak & Roberts, 2022). There are various proofs of his (Fraenkel, 1976, 51), with qualifications (Katz & Katz, 2009, 2010 a, 2010 b), with the devil’s advocate (Richman, 1999), and dissenters (Smith, 1991). Unfortunately, for them, there is a counterargument. That result would likely lead to the mathematicians rejecting the idea that 999… is a real number. So, 999… is both greater than, and less than 1000… The two numbers, 999… and 999…999, can be put into a 1-1 correspondence for every 9 digit in 999… there is a corresponding 9 digit in 999…999, so having the same informational content, they are equivalent.īut, if we take 1000…000, which by (P4) is greater than 999…, and add another infinite string, 899…999, we obtain: ![]() We know, all the digits in 999… are 9’s, unlike with the expansion of an irrational number Ö2, or a transcendental number such as p or e. From the standpoint of mathematical conventionalism (Schroeder, 2018), we have the freedom to define what mathematical concepts we like, how we like, for mathematics is a human symbolic construction governed by pragmatism, not some sort of extra-physical body of eternal truths.Īs one of us argued in 1991 (Smith, 1991), the number 999… could be represented by a quasi-terminating notation: ![]() From these standpoints, a number such as 999 … is just a string of marks, defined by rules as in a “game” (Goodman & Quine, 1947 Weir, 2010). How can this be done, given that the right-hand side is “infinite” and does not have a terminating decimal, although if it did “terminate” in some sense, it would be in a 9 digit, for that is what makes up the number? While this may seem an insuperable difficulty from the standpoint of mathematical Platonism/realism, i.e., the idea that mathematical entities and structures exist in some extra-physical reality as abstract objects/structures, it may not be such a great problem for other philosophies of mathematics, such as nominalism and formalism. This number has some interesting properties. Suppose for the moment that the number 999 … is mathematically legitimate. The proof of the inconsistency of the real numbers is based upon use of a number consisting of a countably infinite concatenations of 9’s: As the latter option is taken to be unacceptable for most philosophers, except perhaps for some paraconsistent logicians, who thrive on paradox, supertasks must therefore be rejected. We show that philosophers of mathematics who accept the conceptual coherence of the idea of supertasks are faced with the proof of the inconsistency of the real numbers. Indeed, some such philosophical assumption is employed in most contemporary “solutions” to Zeno’s paradoxes. The core piece of philosophy involves supertasks, which most philosophers of mathematics accept as at least conceptually coherent, if not metaphysically possible: to perform a countably infinite number of tasks in a finite time. We take no position here on the question of the inconsistency of the natural numbers, but will argue that given certain philosophical assumptions, it can be proven that the real numbers R are inconsistent, yielding 1 = 0. Priest says that he will make no attempt to give an answer of the form “n is so and so,” and “or do I think that an illuminating answer is likely to be forthcoming” (Priest, 1994: p. He does not prove that this is so, which would be a result of some interest, although some have claimed its truth (Zhu et al., 2008). Paraconsistent logician Graham Priest has speculated that the natural numbers N could be inconsistent, such that for some natural number n, n = n + 1 (subtracting n from both sides of the equation yields, 1 = 0) (Priest, 1994). pdf of the complete text of this essay by scrolling down to the bottom of this post and clicking on the Download tab.Īgainst the Philosophers of Mathematics– Either Supertasks, Or the Consistency of the Real Numbers: Choose! You can also download and read or share a. Zeno’s “dichotomy” paradox of motion, (supposedly) solved by completing a supertask
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