Mathematical Reformation Mathematical Speech

1.1) Mathematics consists of a language of the constant relations of positional names.
1.2) By limiting ourselves to the use of positional names, we eliminate from our consideration all dimensions of reality other than the “zeroth” (point), and in doing so provide ourselves with scale independence, with the preservation of position (order).
1.3) The generation of positional names is always limited to a single dimension, expressing relations in n-dimensions, without known closure, because relations of any unclosed set of names cannot be closed.

2.1) Ordinary Speech consists of the language of constant relations between sensory relations.
2.2) The generation of sensory names (nouns and verbs) is limited to the marginal perceivable difference between senses,
2.3) So when we are speaking of mathematics versus ordinary language we are comparing a single dimension for the expression of unlimited relations, and a sensory-limited set of dimensions for the expression of unlimited relations. The principle difference between the two is the precision of positional names versus the information density of ordinary language, and the human ability to intuit the constant relations between multiple sensory dimensions, and our inability to intuit the constant relations between multiple (or even very complex) dimensions of positional names. Our ability to intuit relations between positional names ends somewhere between the second and fourth dimensions. Our ability to intuit relations between sensory names certainly appears to be nearly infinite.

3.1) Now, it is possible to determine and assign all marginally different (perceivable) differences in perception a value in a dimension (number) just as Goedel assigned numbers to symbols. However, the ability of the human mind to separate the sensory-dimensional constant relations of ordinary language(semantics) from the declarative dimensions of positional names is eliminated.
3.2) I would expect in the near future (within my lifetime) I will, or some other person will, develop a scheme of positional names that are a substitute for the sensory names, and that all (english) language will be expressed in such “positional grammar”, and that this will function as an auditable language of artificial intelligence – and at this point it will become possible for the next great transformation of our understanding of the world.

4.1) Wildberger answered the important question of metaphysics very quickly, in that his description of mathematics refers to *computation*.(The grammar of “Action”/Measurment, not of “Meaning”/Fiction )
4.2) An act of ‘writing down’ (speaking) brings a mathematical phrase (description of relations between positional names) into existence, just as speaking a series of phonemes in a sentence brings a description of sensory names into existence.
4.3) He states (correctly) that Computation will eventually “Swamp” platonism (what he call’s rebuilding mathematics).
4.4) He does not state (that I have heard) that he is (like me) restoring Operationalism(aristotelian descriptions), and repairing the re-platonism of mathematics) – restoring the enlightenment(science) from the many counter-enlightenments (fictionalisms).
4.5) Ergo, wildberger speaks of the aristotelian science of mathematics, rather than the platonic fiction (philosophy) of mathematics.

5.1) Mathematics consists of nothing other than the reduction of ordinary language to the grammar (semantics, phonetics, morphology, and syntax) of zero dimensions: positional names. (Assuming we correct the definition of Grammar to include semantics, and redefine semantics as the set of sensory dimensions available to human perception, and therefore constrained by the dimensions included in whatever grammar we use.)
5.2) The failure of the intuitionistic movements in economics (mises), various areas of mathematics (brouwer, poincare, mandelbrot), physics (bridgman), succeeded only in the physical sciences. And the re-platonization of mathematics that had nearly been circumvented by the german second, truncated scientific revolution, was brought into being by the probabilists, including but not limited to Cantor and only countered by The boole, turing, and chomsky, cognitive science revolution, while the philosophical attempt of analytic philosophy from Wittgenstein to Kripke, and the mathematical platonism of Cantor to the present, turns out to have been a rather exhaustive waste of time. The conflation of the techniques of legal, scriptural, hermeneutic interpretation (language) – all of which are justificationary (false) and of sensory scale – to mathematics, which is purely constructive and of single positional scale (the science of measurement by the constant relation of physical names).

6.1) Reduction of ratios (incommensurable relations) to decimal notation (commensurable relations) provides relative commensurability – but at the cost of information loss for any and all non-terminating ratios in decimal notation. This information loss constitutes the primary philosophical problem with the use of ‘infinity’, in mathematics.
6.2) Infinity refers to the consequence of scale independence (absence of limits provided by context), which restored upon application of the description to any measurement of any phenomenon of any scale.
6.3) It is impossible for humans to act (or conceive) outside of actionable scales without entering into the domain of fiction.
6.4) As far as we know we cannot define any existentially possible infinities. Only those things scales too great (or small) as to be inactionable.
6.5) So it is not that any infinity exists, only that the decimal point in decimal notation requires application to scale in other to determine a limit.
6.6) And therefore infinity refers to the preservation of semantic utility (commensurability) and computability at unlimited scales relative to human scale of action, by the use of scaling through the variation in the position of the decimal point in decimal notation.

Curt Doolittle
The Propertarian Institute
Kiev, Ukraine

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