On the Ideal Objectivity of Mathematical Symbols

Is not the ideal objectivity so habitually attributed to mathematics grounded upon a naive foreclosure of sensualism? This shall be our question. This attribution is older than Plato’s view on mathemes; it has a long history.

Introductory remarks:

With ‘matheme’ we understand whatever relevant symbol in whatever mathematical statement. Thus the mathematical statement ’m=e/c2′ consists of a string of mathemes: ‘m,’ ‘=,’ ‘e,’ ‘/,’ ‘c,’ ‘2,’ but also ‘e/c2,’ ‘c2,’ and even the statement itself ‘m=e/c2.’

Mathematics consists solely of mathemes. Mathematics tolerates thus no contagion from semiotics, linguistics, semantics. Mathematics is defined by the very purity of its mathemes. For mathematics to constitute itself, it must allow no relation to language in general, it defines itself de jure and de facto by such exclusion.

There are two classes of mathemes: values and operators. In mathematics there are only symbols that unequivocally are mathematical in nature, thus mathematics is a system of mathemes. Since values do nothing by themselves, mathematics needs mathemes that operate the values: operators.

All mathemes are unequivocally determinable, that is to say they state pure exactitude. Exactitude obtains if and only if mathemes are unequivocally operationalizable and quantifiable.

Only what yields to criteria and procedures of quantification and operationalization determines self-evidential truth; any manipulation of any set of symbols not able to reveal itself as a particular self-evidential truth is therefore not of mathematics.

Mathematics is essentially decisive: its statements are absolute and tolerates thus no relativity, no indecision, no undecidability, no approximation. All mathemes, be they values or operators, must be universally accessible throughout any constellation of historical spatiotemporality. Thus beyond the babel of translation and other idiosyncrasies. Since mathematics thus is decisive it is from its inception the pure manifestation of univocity.

By way of its claim to pure univocity, mathematics is obliged to enlarge its voice till it perfectly parallels the domain of truth in question. Mathematics is therefore a voice that grows to totally adumbration of that truth.

If a set of mathemes consisting of values and operators is to constitute mathematics proper and to hold absolute validity, which is to say hold universally, irrespective of any concrete and historical unfolding of spatiotemporality, it must hold true for all cases for which the statement under consideration holds to be true of. Whatever the specific historical spatiotemporal constellation there must exist kinds of phenomena to which mathematical theories univocally, universally, and self-evidentially applies.

Mathematics is thus supraspatiotemporal. Mathematics is, per definition, transcending any entity whose nature would essentially be relative to history, eventuality, accidence, empiricity, etc. Mathematics is thus not worldly; it is beyond all of the world’s spatiotemporal instances, and herein lies, of course, its Platonism. Now, mathematics maintains that precisely because it is extra-terrestrial, it is valid for all phenomena of the terra firma. The supra of mathematics is precisely what allows it to work in the very internals, the intra, of what it transgresses. Such is the paradoxical subsistence of mathematics. But not so for mathematics itself.

Let us proceed, first, to analyze what is referred as ‘ideal objectivity.’ What are the conditions of possibility for objectivity becoming ideal, thus sur-real? And what are the conditions of possibility as soon as sur-reality is obtained, at some point in an eventual worldly history, for the pure historicity of the traditionalizing of this sur-real of mathematics? Traditionalizing is to be understood as the sole means to hinder the obtained ideality of objectivity from disappearing into the abyss of oblivion. What guarantees a smooth, seamless flow from the infinite archē of the ideal objectivity of the matheme till its infinite telos, then?

Second, we shall have to analyze the hypothesis according to which it is exclusively the supraspatiotemporal nature of mathematics that gives it its intraspatiotemporal efficiency.

Third, we oblige ourselves to question, abiding criteria of scientificity in general, the credibility of such a philosophy of mathematics. If mathematics can lay claim to it heralding an absolute position of the realm of the ideal objectivity—which already is nothing less than dubious post Gödel’s Incompleteness Theorem and other such demonstrations—, what are its merits in the very world it claims to universally and sur-really know and express? Are these in-worldly merits just as unequivocal as its claims to merits on paperpace Gödel etc.?

Fourth, we shall have occasion to read Derrida with a view to clarify what, on the basis of his entire oeuvre, can be said of mathematics, and what can not be said thus. Our working hypothesis will be: mathematics is deconstructable. If, as according to Derrida, the only thing not deconstructible is Justice, then certainly it befalls us to undertake a violent sollicitation of mathematics, no less than is the case with science, or for that matter, no less than is the case with philosophy such as it already is outlined by Derrida himself throughout his entire work.

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Ideal objectivity, thus. What is it? Who, or what, holds it, who or what represents the truth such as it is in and of itself?

According to standard mathematicical doctrine—irrespective of the differences of the schools of logicism (Leibniz, Frege), intuitionism (Brouwer), formalism (Hilbert) and predicativism (Russell)—there would be, paralleled to a certain classical structuralism, a class of ‘mathefieds’ and a class of ‘mathefiers’: any matheme, therefore, consists necessarily of a mathefied and a mathefier. But with certain critical differences, differences that a Lévi-Strauss could only dream of relative the human and social sciences; the structuralist purity that mathematics aspires to can never be obtained in the human and social sciences, since these latter are inextricably bound to linguistic symbols.

The ‘mathefied’ is what is intended to, and what is intended is obviously ideal objectivities that constitute Truth. The mathefied is a pointer from here to a certain there. What is intended is, in the realm of mathematics, always of ideal objectivity; for truth to be truth, and not something else, it must be objective, and this objectivity must be ideal; truth is something that completely annuls, annihilates, any perspectivism, any relativity, any conditionality, and it is essential that this objectivism is without any relation to the real, which is why Husserl will stress the irreal (irreell) character of objectivity. Truth is irreal, and this field of the irreal is constituted by interconnected sets of the matheme qua mathefieds.

The mathefied is thus an element, this or that element, of this vast realm of sur-real truth named ideal objectivity qua mathematics. The ‘mathefier,’ however, is the conveyor of the intended ideal object. The irreducible fact of us living scattered in time and space requires the truth of mathematics to be conveyed to one another. There must be communication, thus. Through mathefiers some part of the total Mathefied, being One, is communicated. Communication produces a sure gradual disclosure of the field of Truth. And it is the mathefiers, analogous to signifiers, that constitute the means of communicating and disclosing Truth. In and through us Truth gathers itself, commencing from a field of scattered subjects, which moreover is bound to remain scattered over time and space, that gradually produces the language of the matheme that is procured by exact procedures of operationalization and quantification.

The metaphysical quintessential presumption of mathematics is that Being conceals an innermost and absolutely self-identical kernel whose nature is such that our cognitive apparatuses, as they are per se (factual or virtual), gains access thereto through mathemes. As if—’as if’ since what here denoted is far from self-evidential—there exists an eternally and universally self-identical kernel of Being, irreducible to and irrespective of any possible spatiotemporal constellation, to be unlocked exclusively by systems of mathemes observing the strictest possible criteria of univocity, decidability, determinability, universality, self-evidence.

Yet more metaphysical presumptions follow, amongst whom the deduction that Being is split dichotomically is the most salient. From which in turn follows the hypothesis according to which whatever does not comply to the form of the matheme must be logically deduced as simulatory, secondary, skin, derivative, external, secondary, arbitrary, false, less, deceptive, etc.—here the lexicon is truly immense—, which is to say not of the truth. Thus the worldly constitutes the non-true, the great deception. Which is why Descartes’ Evil Dæmon—according to which everything that phenomenalizes for us could always be the work of a deceptive, malicious genius—still terrorizes our imagination.

Immediately we see the need to bridge the gap: why is it that, after all, mathemes do exist and are accessible to us, here, in this deceptive quasi-reality? More metaphysics required. In a sense which is not completely without its raison d’être, one could venture to claim that canonized philosophy articulates such bridges, such alignments, such speculations, articulates a certain suture of this abyssal jaw, this khora of chaos. Philodicé abruptly leads to theodicé. And so the history of the et cetera. And the sponte sua. The suture of the khora necessarily involves the et cetera and the sponte sua: go on and on across the gap, all by yourself. There is a certain machinic repetition compulsion to this, but as long as it arises spontaneously it is as if the compulsive is annulled.

In a certain determinate sense, the hypothesis of the evil dæmon still rules our thought, and pulling its strings making thought a marionette: the only way to make the sterile and static matheme appear more verisimilar than the real itself, is to posit that this real is less verisimilar than the system of the matheme. To make the matheme appear more trustworthy than the real and the phuseme, these latter are to be denounced as deceptive in their very nature; only an evil dæmon could make this deception perdure, and so the figure of the evil dæmon will suddenly take on full import.

Descartes’ narration has it that the real is not trustworthy, but that there is a way to counterfeign the untrustworthy and feigned real: mathematics is foolproof. This performative act—once sufficiently upheld to suddenly constitute a pillar in our shared sensus communis—will only gain credibility as the magnificent efficiency of mathemes operating in the real becomes plain to all, at which point the very fact of the history of this particular performative act is relegated to some virtual memory and soon taking the form of being qua obeying the mathematical: what Plato already had suggested, already programming a manichaeism that soon should consolidate the foothold upon which St. Augustine and Descartes could operate their discourses. Alfred North Whitehead’s statement that the Western philosophical tradition are but footnotes to Plato is profound.

Knowledge, or episteme, thus incorporates utter and at first utterly unjustified violence as its condition of possibility: one acts out a performative that simply posits the real as less real and true than our own historical invention of mathemes. As late as with Kant mathematics was held to constitute a parallel world, a perfect intelligible world only awaiting its discovery, from which moment it would only unfold according to the machinery of a priori syntheses. Later, as we gain confidence in the automobile efficiency of mathematics we will, as with Husserl, discard as untenable metaphysics everything that appears superfluous to the successful unfolding of the spectacle of mathematics. Thus Kant’s formal a priori is substituted by Husserl’s contingent or material a priori: mathematics involves a priori truths, of course, but now we do not need it to have been sponte sua constituted as a totally independent realm of truth that awaited its discovery; now the performative maintains that mathematics is a historical invention, by man, whose status is one of pure historical contingency. But one retains the essential core: mathematics is still a science of the a priori, of self-evident truths that procedures of deductions secure the access. Still later we tend towards discarding everything but the sure appraisal of the workings of mathematics; we do not need the narration: mathematics is something that works, and it will expand our understanding until the day of the establishment of the grand theory of everything, at which point, of course, we will not even need the conception of the a priori. At which point a historical performative act has become, not only common sense, but as natural as nature itself, at which point is reproduced in actual time and space creations at will all of which are to be affirmed as of the good.

Also: if the Total Mathefied qua truth is precisely not the sensual qua there-for-us, but whose organization it nonetheless structures, for some reason always imperfectly with empirical flaws, and if homo sapiens however immersed in sensual dissemination nevertheless is per cognitive categories able through exact use of mathefiers to access the truth of math and the math of truth, then implied is that homo sapiens is not of this world, not of nature: the relation of mathematical man and sensual nature is thus irradically allocratic. (Allos: the strange(r), the other(s), the foreign, etc..) But if homo sapiens is constituted allocratically towards nature, and if homo sapiens intervenes in nature through mathemes, what can be expected to result? The only hypothesis that sustains the belief that commanding nature by way of operations of mathemes is on par and in line with nature as such, is that however allocratic the mathemes are, and must be, are they to constitute the pure plane of the Mathefied, there is a certain homonymy or homonomy between sensual nature and mathemes. Because if it was argued that the allocratic relation of mathemes and phusemes (bits of nature) was irreducible and thus a real difference, then one would also be obliged to ponder the significance of this systematic insertion of something wholly other and allos into nature and its phusemes. One must, indeed, in either case, ask: if the founding principle of mathematics maintains that there obtains a relationship of allocracy between mathemes (purity, coherence, intelligibility, etc.) and phusemes (fuzziness, imperfection, eventuality, etc.), then what is the degree of being Other, that is to say if there be degrees? If there is an absolute allocracy, then what sanctions mathematical efficiency in nature? If there is a relative allocracy, what effects will spring forth from the difference of degree? The standard response is that mathemes both takes part of phusemes and constitute something essentially different from phusemes, to wit the mathematical relation to nature is one of relative allocracy. In this way mathematics gains sovereign command over being, in terms of practical efficiency and in terms of absolute indemnity and absolute command.

But such is what is implied in the very conception of the matheme; the matheme, to constitute a proper matheme—and the performative aspect wagers everything—puts itself under uncompromisable duty to posit itself as of in absolute command and efficiency. What do the phusemes state, then? Now that is a wholly another matter. To judge from current states of ecohistory, it does not bode well for the rein of the matheme as hitherto understood. The effect-history of the matheme started out astonishingly, which explains, from the vantage point of the more learned and experienced posteriority, the incredible naivety of the era of Newton and Kant. What Newton accomplished in the realm of phusemes, Kant did with his philosophemes, according to which knowledge, truth, sense, and meaning simply was possible. Newton demonstrated, Kant described, hight of Enlightenment. But if the effect-history of the matheme started out impressive, we are now forced to bracket our superlatives. In fact, judging from the response of phusemes, the mathemes are not unproblematical and without enduring disturbancy and dysfunctionality.

But no, science is in no measure exempt from this cognitive predicament we already alluded to above. Science is de jure and de facto even more immersed in this speculative metaphysics than philosophy ever could be. If philosophy is of “words,” science is of “action.” With effects all the more in the real. When Derrida bespeaks of a necessary counterviolent sollicitation to the equally necessary violence implicit in any philosopheme, here is bespoken a necessary counterviolent sollicitation of the equally necessary violence implicit in any scientific action. Genesis, depicting a creator that creates at will stating all created was created good, lurks in any classical system of the philosopheme, scienceme, or matheme: thus what we call classicism in any of these realms operate a fundamental presumption—once a performative act soon hypostatized as was it evidence itself—according to which whatever homo sapiens create it will be of the good. The era of Newton and Kant could credibly accord its inventions and discoveries such status. Two centuries having passed, we reiterate such acclamation, but with less and less credibility, since an increasing degree of our own inventions and discoveries reveal disturbing signs of not being in naive sync with being and its phusemes.

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