A Math That Includes The Mathematician

I have a color system for the benefit of my readers.

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Science has long excluded the scientist in the name of objectivity. This exclusion was helpful to a point, but it gave rise to a “blind spot”¹. Thankfully, this blind spot is increasingly well tended to in most domains of knowledge…

All domains except for mathematics.

Math is thought to be not only the most objective language available, but a language which is fundamentally objective—a language which couldn’t possibly refer to subjectivity.

I think that’s wrong.

Just as the Subject (and its subjectivity) is now being incorporated into the sciences to form more a comprehensive science (think quantum mechanics, the observer effect, consciousness research), the Subject can be incorporated into mathematics to form a more comprehensive mathematics².

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Mathematics doesn’t include the Subject… or does it.

A more rigorously metaphysical look at mathematics reveals that it does include the Subject—only one has to know what to look for and understand the metaphysics of Subject formation in the first place.

Mathematics fails to include the subject because it does not properly consider its own production of computational indeterminacies.

A comprehensive math—what I would call a transdeterminate math—is one which not only treats certain instances of indeterminacies as proper mathematical objects, but which relates those objects to the formation of the Subject.

Undefined points in mathematics are not voids—they are thresholds of saturation.

(To grok saturation better, refer to my 🟩 introduction to FreQ Theory.)

In traditional math, a function is typically undefined when it encounters a breakdown—division by zero, divergent limits, singularities. But these breakdowns are not emptiness. They are excess: the point at which relational differentiation collapses into compression.

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Where math collapses, the subject begins.

The subject is not something introduced from outside the system—it is what appears when relation becomes saturated beyond further articulation. At these saturation points, a function no longer maps distinct inputs to distinct outputs—the space of distinction collapses. As such, the undefined is the mark of an indivisible coherence—a unity that resists quantification, not because it is mystical, but because it is ontologically prior to relation.

Subjectivity is the coherence of computational excess: where compression becomes so extreme that difference itself folds into unity. Thus a trandeterminate mathematics gestures beyond itself not by virtue of its incompleteness, but by the logic of its own internal intensities.

Keeping the above in mind:

• Computational indeterminacy (incalculability) affords unity.

  • Take this literally. Again, it may help to refer to my soft introduction to FreQ Theory.

    • To briefly explain: Any boundaried system only becomes a unity when the (determinate, particular) objects within it become relationally unified (saturation). Once relationally unified, these determinate objects within the system can no longer be individually calculated, thus rendering the the system indeterminate.

• Unity affords interiority.

  • Meaning the development of RELATION ITSELF, vis a vis the interior/exterior divide, depends on the development of unities.

    • And remember, unities depend on the computational constraints which produced their own internal indeterminacies.

• Interiority affords the Subject.

  • Once an interiority (indeterminate interior) establishes the interior/exterior divide, you have the Subject/Object relation.

We then view the undefined not as an exception but a portal. The following examples aren’t absences of meaning; they are surpluses of relational tension:

• 1/x → ∞ as x → 0, but is undefined at zero.

• tan(x) → undefined at π/2 due to vertical asymptote.

• Divergent integrals and non-converging series mark points of explosive saturation.

Modern mathematics attempts to contain these saturations by formal expansion—but the deeper revelation is ontological.

Mathematical techniques like:

• Compactification (Riemann sphere: ∞ as a point)

• Blow-up techniques (replacing singularities with structured local spaces)

• Renormalization (redefining divergences in physics)

• Distributions (Dirac delta function)

—don’t resolve the breakdowns so much as reframe them.

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…But beneath the reframing is something more fundamental:

These points do not just demand reinterpretation—they reveal the irreducible unity of relation itself, a unity that can no longer be distributed or decomposed. The mathematical singularity is not a defect—it is the site where the subject-object distinction is generated.

These points show us not the failure of mathematical meaning, but the birthplace of relation itself. To the extent that the subject is the unity of relational saturation, mathematics—when pushed to its limits—becomes a metaphysics of the subject.

1. (see here

Matthew David Segall

and

Gregg Henriques

discussing).

2. Let’s not forget it was philosophers (Kant, Hegel), not scientists per se, who first pointed out the problem with a science which avoids the scientist—and it was over a century after their objections that science began to affirm them.