Divorcing Math from Truth
From an engineering perspective, math presents itself as a practical device, a set of tools to be used as tools. Similarly, a physicist uses math to construct models of reality, and as such may be inclined to see the correspondence between math and reality as evidence of mathematical realism. For both the engineer and the physicist, the study of mathematics appears to be a study of the very structure of reality and truth itself, a notion not uncommon to mathematicians who style themselves as practitioners of the purest science, a science upstream from the rest. This is, in truth, a backwards and fanciful idea. Insofar as math actually means anything, it is truly downstream from physics and engineering, and the inversion of this relationship has confused the modern notion of “truth”.
Math is a human contrivance and its correspondence to physical reality only arises because it was meant to. The unquestioned materialist philosophy that underpins the modern science renders the teleology of math invisible, and the answer to the question of whether math is made or discovered is by default taken to be that it is discovered. This is an unconscious assumption that is primarily made because mathematical innovation feels like discovery. The mind is wowed and bewildered by how well math works as a model for physical systems, and such coherence inclines one to believe that something profound has been discovered. But there is nothing truly profound going on here, any amazement ought to fall away once it is understood that math is designed to do this.
In general, there are three ways that math advances: derivation, generalization, and contrivance. Derivation begins with a set of accepted mathematical propositions and, from the implication of those propositions, develops new propositions that are consistent with the first ones. Derivation is therefore math that moves forwards, and generalization, by contrast, is math that moves backwards. Generalization seeks to develop constructs from which distinct but ostensibly related mathematical truths can be derived. The guiding belief behind this practice is that the totality of math can be reduced to a set of axioms from which the rest may be developed. If generalization is the grafting of disjoint branches of mathematics to a unified tree, contrivance might be considered to be the fabrication of such branches in the first place. Contrivance is the fun, speculative kind of math that asks questions like “what if we had fractional derivatives?” or “what would negative dimensions imply?”
Contrivance is difficult to separate from generalization because it generally appears as a synthesis of existing mathematical ideas. Moreover, mathematical ideas are generally contrived with special thought given to how they could be generalized, i.e. they are defined in such a way as to be valid according to existing mathematics. One could also argue that generalizations are just contrivances that allow for different branches of mathematics to be unified. In the context of this discussion, the distinction between contrivance and generalization is made to make clear central idea: math has been contrived to fit the form of physical reality, and this is too often forgotten.
The history of math is full of instances in which a mathematical model is contrived to describe physical reality, and then generalized into broader mathematics and used as a launch point to derive new math. Math has also been subject to natural selection, whereby random contrivances of no initial value become significant branches of mathematics once they are recognized for their value in modelling some kind of physical system. Speaking loosely, the generalization of mathematical concepts that correspond to physical reality backpropagates such correspondence into more fundamental mathematical concepts. It should therefore be a completely expected result that math would correspond to physical reality.
Many of the “immortal” giants of mathematics were physicists and engineers, including Archimedes, Euler, Gauss, Leibniz, Newton, Laplace, Lagrance, Cauchy, Fourier, and Poincaré. Is it any surprise at all that we should see correspondence between the mathematical systems they developed and the physical systems they were developed to describe? The correspondence of math to the material world is one that was intended, and the only thing that ought to be inferred from the result is that mathematicians did a good job.
Euclidean geometry is particularly striking as the most explicitly empirical field of mathematics. In Euclid’s time, geometric proofs were done visually with a compass and straight-edge. Of course, abstraction and generalization via external reflection is still necessary, but correspondence to physical reality was utterly necessity by the nature of the proofs themselves. One might also point to negative numbers, which were for a long time rejected on the basis that there was no rational way to reconcile them with a logical notion of quantification. It was for their use in describing dynamics that they came to be accepted, and a similar story played out with so-called “imaginary” numbers. Infinitesimals saw similar controversy, and arguably the disputes over the veracity of integral and differential calculus were only ever “settled” because of the empirical support for Newtonian mechanics. The Fourier series was contrived to describe heat conduction, and the Fourier and Laplace transforms were generalizations designed for engineering. The impulse function found a home in measure theory, not because it was derived, but rather because it was contrived to describe point-like boundary conditions in partial differential equations, a field likewise developed by physicists and engineers, and generalized into broader mathematics. Indeed numerous advancements in topology and linear algebra can be attributed to their use in modelling quantum mechanical systems, and such physical correspondence is taken into the rest of math when these concepts are generalized. So such ideas like group theory or measure theory find themselves at home in mechanics, not by providence, but by human design.
Natural mechanics, physics, is embedded in the soul of mathematics itself.
The Lie of Mathematical Truth
“Mathematicians do not deal in objects, but in relations between objects; thus, they are free to replace some objects by others so long as the relations remain unchanged. Content to them is irrelevant; they are interested in form only.” -Henri Poincaré
Even if one is ready to admit that certain branches of mathematics come downstream from physics, they might argue that pure math is the logic and truth of reality held in its pure distilled form, so that even if math is developed by and for physics, math in its purity is a truth that precedes and engenders physics. The reasons why math and formal logic are insufficient to describe truth of that sort are very thoroughly explored here, and I would highly recommend a read. Suffice to say for now that form cannot be divorced from its content while retaining any truth, since truth is properly found in the whole. Severing form from content, relationships from objects, and calling it “logic”, is the same as severing facts from reality and calling it “truth”. Actual truth is reality, but so-called “truth” in the Boolean sense is forever consigned to contingency, only “true” relative to baseless axioms. The axioms of mathematics are presupposed, adopted without rigor for the sole reason that they intuitively feel true. Mathematics creates a disconnected world of its own, consisting of elaborate proofs and constructs that need not and often do not find any real world analog or ontology. The condition for “truth” in such a world is little more than 1) beginning with any set of axioms that do not contradict and 2) believing and trusting that this means anything at all. Stripped of its fancy facade, mathematical realism is little better (maybe worse) than religiosity.
“The philosophers make still another objection: “What you gain in rigour,” they say, “you lose in objectivity. You can rise toward your logical ideal only by cutting the bonds which attach you to reality. Your science is infallible, but it can only remain so by imprisoning itself in an ivory tower and renouncing all relation with the external world. From this seclusion it must go out when it would attempt the slightest application.” ―Henri Poincaré, The Value of Science
Godel’s incompleteness theorem dismantled the idea that formal logic could ever provide a complete account of truth, and yet science continues to try. Poincaré, who I love to quote, once remarked that mathematicians would rather abandon reality than geometry in order to reconcile a conflict between them. This rings true in todays scientistic landscape, which responds to Godel’s incompleteness theorem by asserting that reality itself must be incomplete. Logical positivists are so attached to the notion that math corresponds to reality that they would rather assert mysticism than forgo formal logic, believing instead that a complete account of truth is simply unknowable. This is no more intelligent than painting a picture of a horse and concluding that horses are flat and made of paint. The prevalence of math in physics was designed, and it is our design that it is incomplete, not the world.
We ought to dispose the notion that math is, epistemically speaking, anything more than a fundamentally incomplete image of reality, a practical model for use in science and engineering. That isn’t to say that it lacks aesthetic and recreational value, and it isn’t to say that mathematics cannot have philosophical value apart from ontology or epistemology, but the structure of the universe should not be understood to be mathematical. Math is incomplete, the universe is not. The conflation of the two stems from the absolute materialism of our time, the consumption of truth and spirituality by an unquestionably all-encompassing “physical reality”. Math models physics, and because physical reality is taken to be the sole truth of the world, math is mistaken as modelling truth in general. Math, despite its artificiality, is then exalted beyond its proper station, severing itself from and placing itself above physics, then proceeding to build uselessly elaborate and spiritually dead formalities for people to get lost in. We presume that all forms of truth are beholden to formal logic, but this is categorically false, and we would improve in wisdom as a species if this decrepit worldview were abandoned.
“A reality completely independent of the spirit that conceives it, sees it, or feels it, is an impossibility. A world so external as that, even if it existed, would be forever inaccessible to us.” -Henri Poincaré