Dec 29, 2019, 12:00 AM
This article is ideological propaganda (which is common here) in favor of mathematical platonism that intentionally or not misrepresents the problem.
This question of whether .999… = 1 is the canon example, and litmus test, of the conflict over the foundations of mathematics between the schools (a) demanding the scientific basis of mathematics (mathematical realism) by Hilbert and (b) the literary (pseudoscientific) basis of mathematics that was reintroduced by Cantor resulting in the catastrophe of mathematics, logic, and even mathematical physics in the twentieth century. So it is not a question of pedagogy but an unsettled conflict over the choice between mathematical realism under which no infinity is operationally impossible, limits always extant in any application, and therefore .999 != 1, versus mathematical platonism dependent upon the law of the excluded middle, under which deductively, one cannot construct a statement in the vocabulary and grammar of mathematics (the logic of positional names) where .999… does not equal 1. This is the battle between realism (science, operational mathematics), and idealism (philosophy, literary mathematics).
For example, Descartes was important because he restored mathematics to geometry (operations) giving us the cartesian model, and the result was newton-liebnitz’s calculus on one end and the restoration of the realism on the other. Cantor, Bohr, and yes, even Einstein as well as the logicians tried to restore idealism. This led to the constructivist argument. That argument succeeded in physics and has slowly propagated through the sciences, even, oddly causing the reformation of psychology (although not sociology). Computer science has taken up constructivist mathematics leaving mathematical platonism to the discipline of math. Unfortunately, we are stuck with Einstein-Bohr-Cantor versus Hilbert-Poincare-Turing, and this is one of the profound failings ofthe 20th century.
For example. Numbers exist as names of positions and nothing else. We use positional naming to generate unique names. Positions are ordered but scale independent. All of mathematics consist of functions producing names in the grammar and vocabulary of positional names. Cantor states that we can produce multiple infinities of different sizes. This is a fictionalism (parable). Instead, no infinity is constructible only predictable in imagination. So, in any sequence of operations, different sets will produce new positional names at different rates, such that at any given limit, the sets will differ in sizes. There are no different ‘sizes’ of infinities, only different rates of production of positional (unique) names. Math is full of such parables.
In ethics for example, the litmus test is blackmail: it’s voluntary, it’s an exchange, but why do we react against it? Because it’s an unproductive transfer. In logic it’s whether logic is binary and a rule of inference (true vs false) or ternary and scientific (false, truth candidate, undecidable). In mathematics the litmus test is whether .999… = 1. Under realism, no it doesn’t. Under idealism (Platonism) it does. Science (meaning testimony) imposes a higher standard than idealism (platonism). Platonism remains justificationary and Realism falsificationary.
So when you make the claim the question is pedagogical (error) and that people don’t understand – that’s patently false. It’s that operationalism (realism, science) has a higher standard than platonism (idealism, prose). And under realism .999… cannot possible ever equal 1 since no infinity is operationally possible. Whereas under idealism the standard is lower, because under scale independence, infinity substitutes for the unknown limit, which as a consequence is 1.
The fact that people aren’t pedagogically informed that this debate exists, and persists, and that its origin is between western engineering and geometry, and middle eastern algebra and astrology, leading to western reason and science, versus eastern theology and mysticism – then you begin to understand how important this question is – and why our physicists have been lost in mathematical platonism – and why scientific woo woo is so common, when it’s increasingly likely that mathematics of positions names (points) has most likely reached its limits. And that we have failed to create the next generation of mathematics (shapes, geometries) that would allow us to solve protein foldings and the structure of the universe that results in our observed but unsolvable quantum distributions of probability.