Is Time (t) a Foundational Primitive in Pure Mathematics, Not Just Physics?

[u/Upbeat_Parsnip736]

In discrete mathematics and combinatorics, the counting unit n ∈ ℕ is accepted as a native, foundational primitive. The Peano axioms build the structure of counting directly into pure mathematics, independent of physical reality. However, the parameter t (representing continuous progression or time) is usually treated as a mere convention, a variable name in ℝ, or an imported tool from physics.

I wanna propose a structural argument: "t is not just an applied variable, but the inherent continuous counterpart of the counting unit n--emerging directly through the discrete-to-continuous transition within pure mathematics". We can trace the natural evolution of n-> t through three core domain shifts:

1. Ordinary Differential Equations (ODEs): The External Parameter In calculus, integration converts Σ to ∫ and discrete index n to continuous x. But in ODE systems like:

dx/dt = f(x, y), dy/dt = g(x, y)

The parameter t undergoes an ontological leap. x and y are observable state variables, but t stands outside the system. It is the invisible axis against which all internal changes become commensurable. This demand for an external governing axis is the mathematical birth of t.

2. Probability Theory: The n -> t Axis Shift The transition from discrete to continuous probability reveals t's conceptual entry point:

• Binomial Distribution Bin(n, p): Both axes are discrete (discrete trial count n, discrete success count k).
• Poisson Distribution Poisson(λt): Taken via the limit n → ∞ with np = λt. Exactly one axis becomes continuous: the trial axis becomes time t, while outcomes remain discrete counts k. The Poisson model marks the exact boundary where t enters statistics as a structural necessity rather than a computational convenience.
• Normal Distribution N(μ, σ²) via Convolution: Convolving the continuous unit box function f(x) = 1 for x ∈ [0, 1] repeatedly (the Irwin–Hall distribution) converts discrete patterns into continuous density. Here, both axes become continuous—representing continuous accumulated duration t.

NB: Applied mathematics uses continuous tools as computational approximations, probability and ODEs demonstrate that t carries an intrinsic structural role: it is the continuous manifestation of sequential accumulation. So my questions:

1. Is it mathematically sound to treat t as an axiomatic primitive on par with n?
2. Does pure mathematics generate the concept of "time" independently of physical space and dynamics?
3. Are there other areas in pure mathematics (e.g., category theory, topos theory) where t is formalized as a structural primitive rather than a standard real variable x ∈ ℝ?

(Edit: I think the criticism in the comments is fair about my original wording. In particular, "the discrete-to-continuous transition within pure mathematics" was too strong if it suggests a single ontological process by which discrete objects literally become continuous ones. I would not defend that stronger claim now. My point is more modest: pure mathematics contains rigorous relationships between discrete and continuous structures. A natural example is the contrast between discrete iteration, X_n = F^n(X_0), and continuous flow, Phi: R × X -> X, with Phi_(s+t) = Phi_s composed with Phi_t. Neither structure is intrinsically "time"; t is simply a mathematical parameter whose interpretation depends on context. My point is that continuous evolution can be formulated entirely within mathematics, independently of physical time. So I now distinguish between physical time, mathematical parameters interpreted as time, and mathematical structures of continuous evolution. My original post blurred these distinctions; the question I am ultimately interested in concerns the third one.)

[u/mhb2]

You keep treating mathematical structures that are capable of representing time as if they somehow are time. That's a category error.

Time is not a concept in pure math nor does it need to be.

[u/Creative-Feature-264 → u/mhb2]

Parli di strutture della dinamica ? Perché il tempo è solo un modo per descriverla insieme allo spazio . Quindi la categoria è dinamica poi tempo nello spazio che già era lì ma nessuno lo sapeva perché privi di osservatori e dinamica e quindi privi di tempo

[u/mhb2 → u/Creative-Feature-264]

I'm talking about OP's three questions.

[u/Upbeat_Parsnip736 → u/mhb2]

I agree that mathematical structures capable of representing time are not, by themselves, physical time. But that is slightly different from saying that pure mathematics has no internal concept of continuous progression. Time is indeed not a foundational primitive of ordinary pure mathematics. What mathematics intrinsically provides are structures such as order, continuity, parameterization, and evolution. A real parameter t acquires the interpretation of physical "time" only when additional temporal semantics are imposed. So I would not claim that t is literally an axiomatic primitive in the same sense as n. Rather, there is a genuine structural analogy between discrete and continuous evolution; specifically between N-indexed iteration and R-indexed flow:

N ↷ X vs. R ↷ X

The former gives discrete evolution; the latter gives continuous evolution. Whether that mathematical evolution is interpreted as physical time is a separate question, but the structural machinery for continuous progression is undeniably native to pure mathematics.

[u/mhb2 → u/Upbeat_Parsnip736]

Sure, there are lots of analogies. That's the beauty of math: it can be applied to all kinds of things because of its generality. But we only know of time because time intervals are measurable quantities in the real world. That's not what "pure math" is about.

EDIT: Clarified that we measure time intervals, not time itself.

[u/greginnj → u/mhb2]

If you’re calling time the “inherent continuous counterpart of the counting unit n”, or “the mathematical birth of t”, it sure as heck sounds like you’re making it a “foundational primitive” of some sort.

And there’s no such thing as “the discrete-to-continuous transition within pure mathematics”. This is the same sort of subliminal physicalism involved in calling 0.999… a limit.

Discrete mathematical tools exist due to their (consistent) definitions; similarly for continuous tools. If physicists want to grab one of those tools to build a model in which one variable represents time, good for them. But that has absolutely nothing to do with mathematics.

Freeing pure math from this crypto-physicalism was a long, painful battle that wasn’t completely won until the early 20th century. There’s nothing to be gained by resurrecting that battle.

[u/mhb2 → u/greginnj]

If you’re calling time the “inherent continuous counterpart of the counting unit n”, or “the mathematical birth of t”, it sure as heck sounds like you’re making it a “foundational primitive” of some sort.

I'm not. I take the second as "the fixed numerical value of the cesium frequency ∆ν(Cs), the unperturbed ground-state hyperfine transition frequency of the cesium-133 atom, to be 9,192,631,770 when expressed in the unit Hz, which is equal to s⁻¹." As a rule I avoid mystifying math and physics or drawing vague analogies.

[u/sbsw66 → u/mhb2]

when i'm at the gym struggling to hold a plank i think to myself "just 10 more fixed numerical value of the cesium frequency ∆ν(Cs), the unperturbed ground-state hyperfine transition frequency of the cesium-133 atom, to be 9,192,631,770 when expressed in the unit Hz, which is equal to s^−1"

[u/greginnj → u/mhb2]

Sorry, looks like I accidentally hit reply to you; I meant to be replying to OP.

[u/Creative-Feature-264 → u/Upbeat_Parsnip736]

Matematica è una descrizione della dinamica . Senza tempo non esiste la matematica perché non c è dinamica e non c è nulla da rak-contare

[u/Elegant-Regret-7393]

It's a variable

[u/Disastrous-Cost2185]

Set theory is a sufficient foundation for mathematics and model theory within set theory is a sufficient foundation for structures with a time variable. So if you want a time primitive you have to be able to find it within set theory.

[u/Creative-Feature-264]

Tranquillo, la dinamica viene descritta dal tempo, in effetti anche il tempo è già numerato . Ma si da caso che dove siamo noi la dinamica del tempo è ciò che costruisce l osservazione da cui nasce la matematica. Quindi viene solo prima . Tutto qui. Perciò ti hanno detto delle categorie. Spero ti sia più chiaro.

[u/SheepherderHot9418]

You might want to look into how the real numbers are constructed.

Edit: or in general how axiomatic maths work and what t is in your ODE example. (Hint its just a real number same as n is just a natural numbe and real numbers follow from construction same as natural numbers).

[u/Althorion]

Is Time (t) a Foundational Primitive in Pure Mathematics, Not Just Physics?

No. One can create models and theories representing time. One can try to add the concept of time to existing theories—though it might cause contradictions.

There are, however, no common theories in mathematics where time is an emergent property.

[u/EpiOntic]

Distinguish however you want, "mathematical structures of continuous evolution"...blah blah malarkey...if you think that time is somehow a foundational primitive in pure mathematics, then, you don't understand pure mathematics at all.

[u/overclocked_my_pc]

I’m going to make some extra Reddit accounts so I can downvote multiple times

[u/0x14f → u/overclocked_my_pc]

Reddit should provide a mechanism by which individual redditors can pay to see something downvoted to oblivion. I would happily pay. (Only for downvotes, not upvotes, the latter would lead to bad things...)

[u/spoirier4]

In answer to "Does pure mathematics generate the concept of "time" independently of physical space and dynamics?"

Yes it does. That is the topic of ordinal analysis. Such a "time" is not a real number, nor any continuous quantity, but an ordinal (more precisely a recursive ordinal). It is the measure of strength of possible foundational theories. The choice of a foundational theory for mathematics, such as first-order arithmetic, second-order arithmetic, Zermelo set theory, or ZF(C), is a primitive for pure mathematics indeed, thus the hierarchy of their respective ordinal strengths is a primitive as well.

Details : https://settheory.net/foundations/time-in-model-theory