Category: Epistemology and Method

MORE ON MATHEMATICAL PLATONISM

I understand that you can’t convince a theologian to abandon supernatural dependency, nor can you convince a platonist to abandon his independence from reality. Both are habituations where the cost of reformation is beyond their comprehension.

Science depends on realism, naturalism, operationalism, and empiricism is a process of producing testimony in the absence of faith – a warranty to the absence of faith, bias, fictionalism, and deceit. So, there is no substantive difference between the fictionalisms of theology(supernaturalism) and platonism(idealism), including mathematical platonism, where there is a vast substantive difference between Realism and those fictionalisms.

The Realist argument is quite simple: that Realism, including Mathematical Realism, depends not upon declared (arbitrary) axioms but discovered(non arbitrary) laws: Realism, Naturalism, and Operationalism. And Platonism(Idealism) eliminates the dependency on operationalism in exchange for (at the cost of) circular reference, and theology eliminates the dependency on realism and naturalism again in exchange for (cost of) circular reference.

Conversely, to testify to a claim, instead of circular reference we CAN only depend upon the sets: (a) realism, naturalism, operationalism, empiricism, (b) categorical and logical internal consistency, (c) rational choice by known incentives under bounded rationality, and reciprocity by the same criteria in the case of others; and (d) stated limits, full accounting within those limits, and competitive parsimony in between propositions.

So again, mathematics contains many fictionalisms (sophisms) to substitute obscurant non-operational for clear operational causes.

First, the most obvious (as @pennyKarma has stated) is that (i) numbers exist only as names of positions (in an order) and positions in an order alone; (ii) given that all of mathematics is constructed using rational operations (ratio-operations, operations that express ratios) and all of mathematics must be because position is the only constant relation, and (iii) positions produce scale independence, then (iv) a limit is merely the means of arbitrarily choosing the precision at which one rounds upward. So that is step one, the number and rounding.

Step two is the line, and three geometry. For step two the line, there is no number line. There are only positional names. One must fictionalize a line to create dependence upon the line. In other words create a circular reference, a tautology, not a proof.

Step three is the geometric. Let’s take the square root of two which cannot exist (cannot be calculated) without first defining arbitrary limit of precision. The sophistry is that while yes we can deduce a ratio from geometric ideals, the pencil line on paper, or the string used to square four posts in construction, provide limits.

I’ll avoid going through algebra, calculus and statistics on the same premises.

In any event, mathematical sophistry has not gone through the reformation Brouwer recommended, Bridgman achieved in physics, Hilbert warned us about, and Cantor and Bohr buried us in for a century. And while, thanks to Turing, the cognitive scientists forced a reform in psychology by forcing their adoption of operationalism – all too slowly correcting a century of pseudoscience, the Turing revolution and the computer science revolution has failed to inspire a reformation in mathematics – which was lost in sets when it is and must be, like all things, an operational (existential) discipline. Instead, just as philosophers doubled down on the failure of the analytic program, the mathematicians have doubled down on the failure of the pure-mathematics program. So until we discover geometric equivalent of mathematics (which Wolfram has at least touched on), It will be impossible for the discipline of mathematics to reform. And we will continue to see mathematical sophistry and the pseudoscientific nonsense that results from it plague our civilization.

My hope is that de-platonization and de-mystification of mathematics is made possible this century in an effort to improve mathematical education by restoring it to geometric operational and existential rather than set, verbal, and ideal sophistry. But ‘churches reform slowly and only as a last resort”.

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