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Really Small Dicks: Infinitesimals
Infinity Papers Part 6 By Doctor A. Crank


So much then for the infinitely large; there is however even more challenges facing the mathematical skeptic in taking out the infinitely small, infinitesimals. The idea of an infinitesimal quantity played a key role in the development of the calculus, but was regarded as logically inconsistent in its earlier formations. Only in the 20th century, with the use of mathematical logic and algebra, were, allegedly, these difficulties overcome for the notion of infinitesimals. But, I doubt it.

Sir Isaac Newton’s differential calculus, to use Leibniz’s notation, took the “dy” and “dx” in dy/dx to be the ration of infinitesimal differences. The problem facing both Newton and Leibniz’s early formations was that in working out dy/dx, the infinitesimal quantities were assumed to be non-zero in the body of the algebra, or proof, but when we reached the conclusion, to conveniently get the infinitesimals to drop out of the equation, they were assumed to be zero. Critics such as Bishop George Berkeley (1685-1753) and David Hume (1711-1776), argued that this was inconsistent, making infinitesimals both zero and not-zero. See “Continuity and Infinitesimals,” “Stanford Encyclopedia of Philosophy,” at http://stanford.library.usyd.edu.au/archives/fall/2008/entries/continuity/. The problem was taken to have been solved by the replacement of the concept of infinitesimals with that of limits, so that, informally, dy/dx is the limit of the ratio ∆y/∆x, as ∆x tends to 0. However, even the inconsistent theory of the calculus has received a modern rehabilitation by a change from classical logic to paraconsistent logic, which tolerates inconsistencies: C. Mortensen, “Inconsistent Mathematics,” (Kluwer, Dordrecht, 1995); B. Brown and G. Priest, “Chunk and Permeate: A Paraconsistent Inference Strategy; Part I: The Infinitesimal Calculus,” “Journal of Philosophical Logic,” vol. 33, 2004, pp. 379-388. In later papetrs I plan to take the paraconsistency position apart.

Although some mathematicians, such as Georg Cantor, unsuccessfully attempted to prove that infinitesimals were contradictory (see: M. E. Moore, “A Cantorian Argument Against Infinitesimals,” “Synthese,” vol. 133, 2002, pp. 305-330), the 20th century saw the mathematical rehabilitation of the idea of the infinitesimal. The best known account is the non-standard analysis of Abraham Robinson, which will be discussed below.

There are also the “surreal numbers” of J. H. Conway, “On Numbers and Games,” (A. K. Peters, 2001); see also P. Ehrlich,” The Absolute Arithmetic Continuum and the Unification of All Numbers Great and Small,” “Bulletin of Symbolic Logic,” vol. 18, 2012, pp. 1-45. Thus, 1/ω is an infinitesimal, where “ω” is a von Neumann ordinal. When one can just define mathematical entities into existence, what could be easier. There are technical problems with the theory that will be discussed elsewhere: http://mathoverflow.net/questions/29300/whats-wrong-with-the-surreals. The problem of giving a fully general account of integration is apparently not yet solved by surrealist researchers such as Kruskal, but I could be wrong on that one, having limited research opportunities from the lunatic asylum.

E. Nelson also produced infinitesimals by adding axioms to ZFC set theory: E. Nelson, “Internal Set theory: A New Approach to Nonstandard Analysis,” “Bulletin of the American Mathematical Society,” vol. 83, 1977, pp. 1165-1198. This theory is also odd. Consider what J. S. Alper and M. Bridges, “Mathematical Models and Zeno’s Paradoxes,” “Synthese,” vol. 110, 1997, pp. 143-166, say about Nelson’s infinitesimals; “The nonstandard number n can never be constructed because, as Nelson proves, anything that can be explicitly constructed using classical methods is a standard object. In some mystical sense, if it were possible to figure out what n is, then it could not be that.” (p. 152) To my mind, that is an excellent reason for rejecting the theory.

In Smooth Infinitesimal Analysis (SIA), f´(x) is defined as:

f(x+ε)=f(x)+εf´(x). Here, infinitesimals are nilsquare and nilpotent; they are not identical to 0, but their squares are. To carry this piece of intellectual masturbation off, it is necessary to reject the logical law of excluded middle, that pv~p, is logically true. So, x=0 or ~(x=0), for all x, is not logically true: G. Hellman, “Mathematical Pluralism: The Case of Smooth Infinitesimal Analysis” “Journal of Philosophical Logic,” vol. 35, 2006, pp. 621-651. But, don’t ask what x is. If the law of excluded middle held in SIA, then it would be provable that 1=0: see J. L. Bell, “An Invitation to Smooth Infinitesimal Analysis,” at http://publish.uwo.ca/~jbell/invitation%20to%20SIA.pdf.

Along with this, theorems such as the Intermediate Value Theorem, have to be rejected. That useful theorem is: if a continuous function f in an interval [a,b] takes values f(a) and f(b) at each point of the interval [a,b], then it also takes any value between f(a) and f(b) at any point in the interval. One could turn the tables on SIA and argue that these results are grounds for its rejection, and that rejecting the law of excluded middle is ad hoc. Apart from the inconsistency, there is no independent reason for rejecting this principle. Sure, there are alternative logics, but so what? Any logical principle can be rejected, including the law of non-contradiction. If one had the ball, one could even allow absolute inconsistencies, which would enable everything to be proved (and the negation of everything), which would get the whole game over in a day.

The most famous theory of infinitesimals is that of Abraham Robinson’ non-standard analysis: “Non-Standard Analysis,” (Princeton University Press, Princeton, 1996), and found in calculus books such J. Keisler, “Foundations of Infinitesimal Calculus,” (1976). An infinitesimal ð is an element of the non-standard real numbers and is defined for a positive ð as less than r for all real r. An infinite number is greater than all real r. Two elements x and y, are infinitely close if x-y is infinitesimal. A number z is infinite if the inverse of z, 1/z, is infinitesimal. See A. H. Lightstone, “Infinitesimals,” “American Mathematical Monthly,” vol. 79, 1972, pp. 242-251.

Now if all that is needed to present a mathematical idea is a definition, and maybe freedom from absolute inconsistency, then even the sky is not the limit. Thus, we could define the hyper-infinitesimal numbers to be numbers greater than 0, but smaller than any non-standard infinitesimal, and so on for hyper-hyper-…numbers. Robinson, though, attempted to show that non-standard numbers actually existed, whatever that now means.

To do this he constructed non-standard models of the reals, and made use of the model-theoretic version of the compactness theorem: if every finite subset of the set of proper axioms of a first order theory T has a model, then T has a model. A “model” is defined as: an interpretation I is a model for a set of formulas of a theory T,* if and only if it is true for I. One then considers a first-order theory adequate for the real numbers, R. A constant c is added to r and a denumerable infinity of axioms: there exists an x, such that x=c; c>0; c<1/2; c<1/3; … etc. The new system is called R.* By the compactness theorem, every finite subset of the axioms of R* has a model, so R* has a model. From this it is concluded that there is an object in R* which is greater than 0, but less than* any positive real number, and this, allegedly, is an infinitesimal.

Or is it? One mathematical logician, Geoffrey Hunter, “Is Consistency Enough for Existence in Mathematics?” “Analysis,” vol. 48, 1988, pp. 3-5, argued that there are no unique non-standard models, and the models are not isomorphic to each other, so the set of infinitesimals is not well-defined. But, did anybody care?

However, there are other problems: “But was that existential proposition (that infinitesimals exist) established simply by establishing consistency? I think not. At a crucial point you have to show that every finite subset of the axioms of R* has a model. Or else somewhere in proving the Compactness Theorem you have to appeal to a previously proven theorem that certain sets of formulas have models. That is, at some point you have to establish that a consistent set of formulas applies to something.” (p. 5) I think another counter-argument, from a finitist position, is to put them to proof that every – and that is at least a denumerable infinity of subsets of the axioms of R* have models – and that supertask cannot be performed.

Hunter goes on to point out that there are technical results challenging the idea that formal consistency entails having a model: there can be single sentences which are consistent, but not satisfiable due to Gödel’s incompleteness theorem: L. Henkin, “Some Interconnections between Modern Algebra and Mathematical Logic,” “Transactions of the American Mathematical Society,” vol. 74, 1953, pp. 410-427, at p. 425. So, Robinson has not succeeded in proving that his infinitesimals exist.

There are doubts expressed by a minority, well, by Antonio Moreno, “The Calculus and Infinitesimals: A Philosophical Reflection,” “Nature and System,” vol. 1, 1979, pp. 189-201, about the cogency of non-standard analysis. Consider a point (x0,y0) on the curve y=x2. Let dx be non-zero, but a positive or negative infinitesimal, and dy the infinitesimal change in y. The slope of a tangent to the curve at (x0,y0), is defined as:

Slope at (x0,y0)= the real number infinitely close to dy/dx.

Thus, dy/dx=((x0+dx)2 –x02)/dx = 2x0+dx. This is a hyper-real number, but since dx is infinitesimal, 2x0+dx, is infinitely close to 2x0, which is real. Therefore the slope at (x0,y0), is 2x0, the standard result. That looks good, only there is no real number infinitely close to dy/dx, which is used in the definition of the slope at the point, because real numbers simply don’t get infinitely close in this non-standard sense, to anything. If the slope is defined as the real part of the hyper-real number, that is consistent, and works fine. But, there was no need for all of the logical machinery to do this trick, and Leibniz could have got away with the same thing, simply by saying that the slope was the real part of the “number” and the infinitesimals were sui generis, inconsistencies that could be ignored by arbitrary definition.

Some further skeptical arguments are given by H. Slater, “Logic Reformed,” (Peter Lang, Bern, 2002), pp. 171-177. Slater also has an attack on Weierstrass’ definition of the derivative, at least from a finitist position, because it presupposes the notion of a real number. W. Zhu (et al.), “New Berkeley Paradox in the Theory of Limits,” “Kybernetes,” vol. 37, no. 3/ 4, 2008, pp. 474-481, go further, and give an argument purporting to show that even the limit account of Weierstrass still does not escape Berkeley’s paradox.

Conclusion

Infinity in mathematics, logic and physics is an unjustified bucket of shit that needs to be emptied, and emptied soon.

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