Many people think of math as apodictic. That is, a language that is demonstrably, necessarily, or self-evidently true, leaving absolutely no room for doubt or contradiction. Certainly, there are propositions in math that are demonstrably, necessarily, or self-evidently true because the conclusion can be built on the premises. This type of argument is called a syllogism. A syllogism is a form of deductive reasoning where a conclusion is drawn from two given or assumed premises.

• Major Premise: A general statement believed to be true (“All humans are mortal.”)
• Minor Premise: A specific instance or example of the major premise (“Socrates was a human.”)
• Conclusion: The logical outcome derived from combining the two premises (“Therefore, Socrates was mortal.”)

Why It Works

• Validity: This specific argument is logically valid because the conclusion follows necessarily from the premises.
• Soundness: It is also sound because the premises themselves are actually true in reality.

In writing “actually true in reality,” we are referring to the Greek word ontos. The Greek meaning of ontos is something like what we mean by being, existence, that which, we say, actually is. But ontos is always together with ontological. Ontological adds the Greek suffix -logia, meaning logic or, more generally, the study of. Every ontos is necessarily accompanied by a logic, a -logia. If ontos is the physical house, ontological is the blueprint, the architecture, and the very concept of what makes a structure a “home.” It asks questions like: What is the difference between a physical object and its idea? Do numbers exist in the same way trees do? Well, trees can be numbered, but trees are not simply numbers. Likewise, ontology is a way of finding similarities. It seeks the underlying structure—the -logia—that connects disparate things. However, a deductive syllogism is a structural tool, whereas an apodictic proof is a structural guarantee.

• The Difference: A syllogism can be valid but completely false in reality (e.g., “All aliens are green. Yoda is an alien. Therefore, Yoda is green.”). This is a perfect deductive syllogism, but it is not an apodictic proof because the premises are not self-evidently or necessarily true.
• The Connection: A deductive syllogism becomes an apodictic proof only when the premises are absolutely indisputable (like in mathematics: A = B, B = C, therefore A = C). If the premises are necessarily true, the deduction yields an apodictic proof.

This is the boundary between the rules of thinking and the nature of reality.

• Ontos (The Factual Being): As established, ontos is the raw fact of existence. An apodictic proof does not just tell us that something happens to exist (ontic/ontos); it proves that something must exist by logical necessity.
• Ontological (The Framework of Being): An ontological argument tries to use pure, abstract reasoning about the concept of “Being” to reach an apodictic conclusion. For example, classic ontological proofs argue that the very definition of a “supremely perfect being” logically requires that being to exist, making its non-existence a logical impossibility.

So apodictic proof necessarily relies on a narrowing of meaning or, better, an abstraction that relies on a kind of redundancy. For example, the formula I gave about A, B, and C is so generic and abstract that it can carry the weight of an apodictic proof—but what do we sacrifice in that extreme of an abstraction?

The irony of apodictic proof is its cost: to achieve absolute mathematical certainty, we must completely empty our arguments of the real world. Apodictic certainty doesn’t look at reality; it looks at its own frozen definitions. This is one of the most profound critiques of formal logic in the history of philosophy. To achieve the absolute certainty of an apodictic proof, we must strip away reality and retreat into extreme abstraction and redundancy. When we reduce the world to generic formulas like A = B and B = C, we sacrifice three massive elements of existence:

1. We Sacrifice Content for Form (The Loss of Substance)
To make a statement apodictic, it must be true in all possible worlds. The only way to make a statement true in all possible worlds is to empty it of any specific real-world data.

• The Sacrifice: We sacrifice meaningful information about the universe. As the physicist Albert Einstein famously noted: “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.”

2. We Sacrifice Change and Time (The Loss of Becoming)
Apodictic proofs require identities to remain perfectly fixed. A must always equal A.

• The Sacrifice: We sacrifice time, mutation, and evolution. In the real world (the realm of ontos), everything is constantly changing, decaying, or shifting context. A seed becomes a tree; a living human becomes mortal remains. Abstraction forces a fluid reality into frozen, static concepts to make the math work.

3. We Sacrifice the Unique for the Redundant (The Loss of Particularity)
This abstraction relies on a kind of redundancy (or tautology). To say “All humans are mortal; Socrates is human; therefore Socrates is mortal” is really just stating the same definition over and over again. The conclusion is already buried inside the first premise.

• The Sacrifice: We sacrifice singularity. To force Socrates into the abstract category of “human,” we must ignore everything that makes Socrates Socrates—his unique thoughts, his eccentricities, his specific existence. We trade the rich, messy reality of an individual being for a neat, repetitive category. In the realm of pure logic, the particular (ontos) is erased in favor of the universal (logia), but something vital is lost in the exchange.

The problem occurs when the line between what is necessarily true blurs into what is apodictically true. This would be called a logical fallacy. When certainty and academic acclaim (or the pursuit thereof) seep into what we call common sense, we end up with a worldview, a reality view, an absolute or unquestionable belief. Deduction/Apodictic Proof gives us absolute certainty at the cost of erasing the real world. The answer to this dilemma is called inductive logic.

Induction gives us real-world knowledge at the cost of losing absolute certainty. Inductive reasoning is the perfect antidote to the frozen abstraction of apodictic proof because it does the exact opposite: it embraces the messy, changing, and real-world realm of ontos. Instead of starting with an abstract definition and moving down to a fixed conclusion, inductive reasoning starts with real-world observations and moves up to form a general rule.

1. The Core Difference: Probability vs. Certainty

• Deductive/Apodictic: Starts with a cosmic rule (“All humans are mortal”) and guarantees a specific conclusion (“Socrates is mortal”). It offers 100% certainty but tells us nothing new.
• Inductive: Starts with specific data points (“Socrates died, Plato died, every person we have ever observed has died”) and infers a general rule (“Therefore, all humans are likely mortal”).

2. What Induction Reclaims (And What It Sacrifices)
In this way, mathematics and philosophy try to win back what abstraction threw away:

• It gains real-world content: Inductive arguments are entirely about the physical world (ontos). They deal with real things you can see, touch, and measure—like observing thousands of white swans before concluding that “all swans are white.”
• It sacrifices apodictic certainty: Because induction relies on observation, it can never be apodictic. It can only ever be more or less probable. All it takes is a single mutation, a passage of time, or a new discovery (like finding a black swan in Australia) to shatter an inductive conclusion.

So now, the original goal of my post is to show how mathematics 1) functions as a language, 2) has built-in paradigms and worldviews of its period (dare we say metaphysics), 3) as a language receives guarantees of common sense in acceptance (or rejection via paradigm shifts), and 4) accommodates the past of habit and pattern and the future of the novel and promising. However, mathematics does offer a certain amount of shelter from historical ways of thinking (common sense).

in its specialized language, but also a certain amount of delusion in that its language is just as vulnerable to radical changes and shifts in meaning—as evidenced in the history of science—as the metaphysics that science condemns. This is not to say they are the same, but the dynamics of mathematics can be given a sort of sanctity that language is often condemned for.

Mathematics isn’t an eternal, sterile truth dropped from the sky. Instead, it is a living human enterprise—vulnerable to the same shifts, habits, and cultural frameworks as any other language. Mathematics operates as a highly specialized language, but languages are never neutral. Every mathematical era carries the hidden metaphysical worldview of its time. As an example, Ancient Greek mathematics was deeply tied to ontos—the physical reality of geometric space and whole numbers you could touch or draw. They rejected irrational numbers or zero because their worldview couldn’t conceptualize “nothingness” or “infinite chaos” as having being. Their math reflected their metaphysics. We might wonder: while the particulars of our mathematics are not the same, might there be some form of continuity between the “temporally antiquated false assurances” we now derogatorily reference as metaphysics and the veracity of scientific truths? And, at the same time, could language open up new possibilities, where novel and radically different human dynamics (ontologies) may be forerunners to particulars of science (ontic) and their findings (ontological)?

Because math is a language, its survival relies on a community accepting its definitions. What we call “mathematical rigor” can have an unmistakable resemblance to what we often mean by “common sense” or the consensus of a specific historical period. When that consensus fractures under the weight of new crises, a paradigm shift occurs. I discussed these kinds of shifts at length in 20th-century physics with relativity and quantum physics in my books Quanta, Alterity, and Love. In these cases, we can see how language changes to accommodate, to rewrite what is considered “self-evident.” And I might add that in quantum physics, the very notion of a generic self in “self-evident” is made archaic by the non-generic-able observer. More importantly, quantum physics poses a dramatic question to any ontology, any one particular, absolute, classical reality. At the same time, it gives us predictions and possibilities within the context of probabilities while also blowing away any possible single, coherent underlying (ontological) reality (either/or not applicable here). In any case, language and mathematics both retain habits of the past juxtaposed with promises of the future—they temporalize.

In using the word temporalize, I want to give a case in point of what some meanings could be. Relativity makes a single, universal temporality less than the speed of light impossible, and at the same time, it makes the speed of light absolute but without time. What does this mean? Time itself is relative to our perspective—our time perspective actually slows down the closer we see something moving towards the speed of light. We see light or photons traveling through the universe, but photons themselves see no universe in which they are traveling since they travel at the speed of light; e.g., there is no faster or slower for photons. We need to rid ourselves of the idea that light itself has any experience of time. Additionally, quantum entanglement of particles or systems of particles has nothing to do with time at all! We naturally think that from our relative perspective an entangled system on one end of the universe would take time to change its state on the other end of the universe. But entangled systems on opposite sides of the universe take no time to change their state. It is as if space and time do not exist at all; as if they were the same thing or whatever we mean by that.

To go one step further, quantum physics tells us that temporality and what we see or experience is totally unique to the observer and yet still not merely something we make up from scratch. Observation collapsing the wave function (analogous to light traveling through space) may take on probabilities in highly controlled and isolated experiments. But in observing, we have already spawned all possibilities into what we call reality (called the collapse of superposition). So, for Schrödinger the unobserved cat is both alive and dead. This means the act of observation doesn’t just reveal a pre-existing state; it actively participates in bringing specific temporal realities into being from a field of timeless/spaceless potentials that get actualized. So, observing both temporalizes (for us, the observer, and all other paths) and simultaneously remains in superposition as if non-temporal-spatialized in the un-collapsed superposition state—the wave function, which in some spooky way already anticipated/knew the observation and retains no recognition of a collapse at all. Perhaps an analogy might be helpful. We experience faster and slower, while photons traveling at the speed of light have nothing to do with our perspective. Read my book for a more detailed account. Math loses the burden of having to comply with misconceptions of what we think of as ‘conventional’ language but at the same time retains its own histories, assumptions, and archaic weight (the Copenhagen Interpretation) which tend to dance with semi-classical notions as well.

Math is a battlefield between inertia and novelty, and let’s not forget monetary reward (which is much harder to achieve in the marketplace of pure science per se). The past carries the immense weight of historical habit, notation, and rigid axiomatic patterns. The future accommodates the wild, inductive leaps of intuition where a mathematician spots a novel pattern and bets on a promising new conceptual framework before it is ever strictly proven. The power of truth per se is a delusion of the “apodictic.”

As shelter, mathematics uses its highly abstract, generic language to isolate itself from everyday historical biases. It also wants to align itself with logic as I discussed in this recent post: The Heroic and the Religious. Mathematics seeks to retreat into the “redundancy” of pure logic (A = B, B = C, therefore A = C) to find a safe haven of absolute, apodictic certainty, far away from the messy, changing world of ontos. As delusion, the apodictic is the belief that this shelter makes math immune to history. It is just as vulnerable to radical shifts in meaning as any language. Some might argue that science seeks to give us a clear context for predictability, and that is true. However, language itself also tries to give each of us a clear context in our daily lives, beliefs, and values while at the same time giving us a sense of predictability in our decisions. Ah, but you might think science seeks consensus in its goals. But likewise, so does language as government. Furthermore, the goal in common language may not be a formal government, as in a ‘democratic government,’ but it can also provide a governing commonality in informal societies and norms/standards for relationships. This parallel suggests that mathematical ‘certainty’ might function more like a social or governmental ‘consensus’—a historically situated agreement—than an access to timeless truth. Additionally, we should not think of different languages as having some common universal language between them. In his lectures on science and language, Heidegger repeatedly remarks that “translation is always already interpretation.” So, languages like mathematics and ethnic languages have no metaphysical sameness that is simply a matter of different words—same definition. Differences in languages prevent simple equivocations.

Modern science frequently condemns philosophy or metaphysics for being “unverifiable” or “bound by language.” Yet science grants a sort of sacred, unquestionable sanctity to mathematics. The “sanctity” of an apodictic mathematical proof is entirely dependent on a historically bound, human-constructed linguistic framework. When the paradigm shifts, the “eternal truth” shifts with it. At the same time, mathematics’ affinity to mere human languages can and does radically reshape our ontological horizons. Furthermore, I think quantum physics seriously needs to be made more explicit to our common, human-styled languages to make apparent how radically different it is and, more importantly, to help us understand that radical differences are NOT ALL THE SAME. Some differences are merely variations within a known system, while others represent a true break, a genuine “alterity” that challenges the system itself to incomparable. Even the possibility for normalizing all these varieties of radical differences has now been denied to us in the mathematics of the quantum wave function and its many ontologies and non-comparable offsprings—not even to begin to think of radical alterity in the other: the she, he, or pick your pronoun of the one who faces us. Could facing the other be the radical alterity from which we flee into history and ego?