r/PhilosophyofMath

Hi everyone,

I just recently made a video covering the history and development of proof theory in under 1 minute, and I’d really appreciate some honest feedback from this community.

As I am interested in mathematical logic, I’ve always been confused by what seems to be a neglect from the rest of the larger math community. I found that a lot of videos either skip over the history or get too bogged down in formalism. I tried making a good overview for the beginner that doesn't talk down to you.

If this isn't the right type of post for this community, please let me know and I'll move it.

Thank you all for your time.

If an infinite number of proper classes can be formed by infinitely including or excluding sets from the class of all sets V, could that infinite number of proper classes be considered an infinite number of absolute infinities or just an infinite number of ways the only absolute infinity which would be the class of all sets V can be sliced (if of course a proper class can be considered an absolute infinity)?

Every natural number n has an endsegment {n+1, n+2, n+3, ...}. The first endsegments are

{2, 3, 4, 5, 6, 7, ...}

{3, 4, 5, 6, 7, ...}

{4, 5, 6, 7, ...}

...

Some defenders of set theory claim that the intersection of all endsegments is empty while no endsegment is empty. I call this statement matheology, a touchstone of irrationality.

Regards, WM

A consensus is emerging among respected mathematicians that there is a decent chance AI will exceed humans in both brute force verification and complex, creative problem solving at the highest levels. Few frontier theorems will be proven by humans alone, perhaps, in a matter of years.

More controversial, AI may also outpace humans in shaping the correct definitions, building theories, and making connections between disparate areas of mathematics, oft considered the peak of human creativity in mathematics. New areas of mathematics may be created without much human guidance.

Perhaps mathematicians become as helpful to AI as toddlers are to mathematicians. One cannot confidently rule out this scenario - Terrence Tao may find himself completely useless in building a rich, beautiful body of new mathematics.

In such an extreme scenario, humans would still matter in mathematics!

Lockhart's Mathematician's Lament argues for the intrinsic beauty of mathematics as being of primary importance. Humans, as knowledgeable appreciators of beauty, thus play an important a role as spectators and enthusiastic amateurs in mathematics, even if they cannot be world-renowned "competitors" in theorem-proving and theory-building. This mirrors the situation in chess, where the vast, vast majority of human chess players and appreciators will never contribute to the frontier of advanced lines, and arguably even the most skilled like Magnus Carlsen rely on AI to develop their strategies, and would be crushed by such AI in competition. Being completely uncompetitive does not make chess playing and appreciation valueless.

> Amateur: from French amateur "one who loves, lover"

But there is more beyond this. Mathematics allows you to understand things that are otherwise impossible to understand. Some of these are important for fairness and justice: Arrow's impossibility theorem, statistical bias, observer relatively and other tricky concepts around coordinate systems (map != territory), locally-trivial globally-nontrivial (global obstructions), forgetful maps to extract the essential structure and remove irrelevant details, limits of computation, etc.

Understanding such mathematical concepts allows you to make moral judgements in ways that would be impossible otherwise. Some super-smart machine might tell you Arrow's theorem is true, but internalizing it yourself gives you the rich understanding of fairness in democracy necessary to consciously shape it. As with humans surpassing the capabilities of their own eyes with optical then radio telescopes, we are not impoverished by using tools that allow us to extend our reach into things we can never directly perceive or understand.

It can be frightening because the life's work of someone of the previous generation can be reproduced and surpassed flippantly. Gauss himself spent a significant amount of time manually factoring prime numbers by hand, a tedious exercise upon which his conjecture on the distribution of primes (the prime number theorem) was based. Gauss died before his conjecture was proven. His notebooks full of rote calculations could be reproduced today in a fraction of a second so short you could not perceive it. Anyone today repeating an endeavor like Gauss by hand would be thought a fool, just as an astronomer who refuses to use a telescope.

That doesn't make the pursuit of understanding pointless. As the limitations of our use of AI will stem from limitations of our own minds, it will still be profoundly rewarding to practice mathematics. Indeed, we may spend less time performing rote exercises and miring in false conjectures. Already the body of mathematics is too large for any single person to understand. One can pessimistically reduce mathematics to mechanics, or optimistically find meaning in your particular path through the mathematical version of the library of babel. Because we shape our minds, our society, and our world with mathematics, we will always matter as sentient beings who reify mathematics by subjecting ourselves to reason, and better ourselves because of it.

I have seen a lot of posts here, some very interesting. The philosophy of mathematics is naturally an interdisciplinary subject sitting at the crossroads of math and philosophy. I guess people might bring different contributions and perhaps even come to different conclusions depending on whether they're primarily philosophers or mathematicians. Hence the poll.

Feel free to give a more specific answer in the comments.

IMPORTANT NOTE: By academic background I mean some kind of degree in the subject, a published paper or at least having taken a decent chunk of undergrad. If you're only interested/curious but have no formal training, please answer NEITHER / OTHER.

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THE COLLATZ CONJECTURE

A COMPARATIVE DENSITY PROOF OF TRAJECTORY DESCENT IN THE COLLATZ 3N+1 SYSTEM VIA 1N+1 MODULAR MODELING

Author: All mathematical ideas and constructions by Steve Tomlinson except logarithms in 2 and 3.1.

(l knew something mathematical must do this job, l didn't know what it was; logarithmic bounds.)

Essay composition by AI with many mistakes edited by Steve Tomlinson

Date: August 2026

ABSTRACT

This paper establishes a novel structural framework for analyzing the Collatz 3N+1 conjecture by introducing a perfectly descending baseline model: the 1N+1 system. While the standard 3N+1 system exhibits chaotic trajectory growth, we prove that both systems operate on base-2 modular architecture. By comparing the density pathways of the 3N+1 system against the verified, linear geometric descent of the 1N+1 model, we demonstrate that the standard Collatz mapping exhibits an absolute asymptotic density of descent equal to 1 at the infinite operational horizon.

1. THE CENTRAL BREAKTHROUGH:

THE 1N+1 STRUCTURAL BENCHMARK

To analyze the non-linear trajectories of the standard Collatz conjecture, we define a perfectly controlled model system, the 1N+1 system, governed by the following mapping for all natural numbers N:

f(N) = N/2 if N ≡ 0 (mod 2)

f(N) = 1N+1 if N ≡ 1 (mod 2)

Theorem 1.1. In the 1N+1 system, 100% of all natural numbers N > 1 are mathematically guaranteed to reach a strictly smaller value within a maximum of two operations.

Proof.

Case 1: If N is even, a single operation yields N/2, which is strictly less than N.

Case 2: If N is odd, the application of the odd rule followed by the mandatory division by 2 yields a composite operation of (1N+1)/2.

Setting up the inequality for descent:

(N+1)/2 < N => N+1 < 2N => 1 < N.

This inequality holds true for all positive odd integers greater than 1. Thus, every element shrinks locally and immediately.

By creating an arbitrary system for numbers to drop in the 1N+1 system, the entire number line is partitioned into clean, un-scrambled geometric slices:

* Step 1 (All Evens, 0+2n) accounts for exactly 1/2 (50%) of all numbers.

* Step 2 (The 1 + 4n Odds) accounts for exactly 1/4 of all numbers

*Step 3 (The 3 + 8n Odds) accounts for exactly 1/8 of all numbers.

*Step 4 (The 7 + 16n Odds) accounts for exactly 1/16 of all numbers.

Continuence of this process continues to account for exactly (2^x-1)/(2^x) of all numbers, accumulating to 100% of the number line descending within a 2-step horizon.

2. THE 3N+1 SYSTEM AS A LOG-LINEAR DISTORTION

When the odd operator is shifted to the standard Collatz rule (3N+1), the underlying base-2 modular grid is stretched. Let m represent both the family classification and the number of odd steps executed before the first downward drop below the initial value. Let a be the number of required even operations (divisions by 2).

For a net trajectory descent to occur, the geometric growth factor must drop below 1:

(3^m) / (2^a) < 1 => 3^m < 2^a

Taking the base-2 logarithm (log₂) of both sides yields the absolute structural boundary:

a > m · log₂(3) ≈ 1.5849625m

Because log₂(3) > 1, immediate descent within a single operational cycle is impossible for odd positive integers. Instead, numbers are sorted into deterministic "m-families", where the total step horizon required to secure the necessary 'a' divisions scales linearly as a function of m:

Total Steps = m + a = ⌈2.5849625m⌉

3. THE m-FAMILY SIEVE AND EXPONENTIAL CONTRACTION

The exact proportions of the number line accounted for by these families are defined sequentially:

(Instant Evens, 0 + 2n) accounts for exactly 1/2 of all numbers.

* m=1 (The 1 + 4n Odds) accounts for exactly 1/4 of all numbers.

* m=2 (The 3 + 16n Odds) accounts for exactly 1/16 of all numbers.

* m=3 (The 11 + 32n Odds) + (The 23 + 32n Odds) accounts for 1/16 of all numbers.

*m=4 (The 7 + 128n Odds) + (The 15 + 128n Odds) + (59 + 128n Odds) accounts for 3/128 of all numbers.

* m=5 ((The 39, 79, 95, 123, 175 and199) each + 256n Odds)) accounts for exactly 7/256 of all numbers.

* m=6 ((The 287, 347, 367, 423, 507, 575, 583, 735, 815, 923, 975 and 999) each + 1024n Odds) accounts for exactly 12/1024 of all numbers.

At this point when m reaches 6:

(6 × 2.5849626) rounded up = 16 Collatz operations accumulates to account for exactly 15/16 of all numbers shown to reach a smaller number.

* m=7 accounts for exactly 30/2048 of all numbers.

Manually proving m=8 would have taken too much paper.

3.1 The structural limits for m=3 and m=4 families in the Collatz conjecture are determined by the logarithmic boundary

a > m×log2(3), where m is the number of odd steps and a is the number of even operations. Applying this, the m=3 family requires 5 even steps for 3 odd steps, creating a 1/16 density across residues modulo 32, while m=4 requires 7 even steps for 4 odd steps, generating a 3/128 density modulo 128. This logarithmic framework accurately predicts the modular structures for specific families.

4. THE UNIFIED 2^x HORIZON INDUCTION

While the multiplier 3 introduces "bumpy" intermediate statistical fluctuations between the milestones (e.g., stabilizing around a cumulative density of ≈ 5/6 at step 6, 10/11 at step 11, 12/13 at step 13, back to exactly 7/8 at step 8 and 15/16 at 16 steps) The total system mathematically self-corrects and snaps perfectly back to the clean geometric density progression of the 1N+1 benchmark at every power-of-two operational milestone (2^x).

By mathematical induction on the operational horizon x, the cumulative density of numbers proven to have reached a smaller value satisfies:

Cumulative Density(2^x) = 1 - 1/(2^x)

As the operational step horizon scales toward the infinite limit (x → ∞):

Limit as x → ∞ of [1 / 2^x] = 0

5. CONCLUSION

By using the 1N+1 system as an absolute structural baseline, we prove that the standard 3N+1 Collatz system is not chaotic, but deterministic and rigidly bounded. The "numerical shields" created by dense clusters of binary ones (such as the 2^x - 1 Collatz steps families) only temporarily delay descent. Over an infinite horizon, the remaining density of holdout numbers converges to exactly zero.

Because 100% of all numbers must eventually reach a strictly smaller milestone, any arbitrary starting number is locked into an inescapable cascading chain of downward thresholds, forcing all trajectories to eventually collapse into the fundamental 2 → 1 trivial loop.

Q.E.D.

By shifting the analytic paradigm from stochastic modeling to comparative structural architecture, this 1N+1 baseline framework introduces a constructive element that establishes absolute structural determinism, distinguishing it from the probabilistic approach in Terence Tao’s 2019 groundbreaking density proof. While Tao’s work treats individual trajectories as non-constructive, semi-chaotic random walks, this model maps the geometric architecture of "numerical shields," demonstrating that standard Collatz mapping is rigidly constrained by a base-2 modular grid and logarithmic boundaries.

One is 0% in infinity, and therefore, one is nothing at all. Unless, of course, there is an end to infinity. One number contains an infinite number of decimals, although in an oscillating infinity, one number cannot go on forever, as other numbers are included in that same space, for that infinitely existing number has bounds. Can the same thing be said about the universe? Would that mean there are infinite infinite universes? And, as even further food for thought, could the same thing be said for individual minds?

Idk, I had a shit ton of Monster Energy just now and my account isn't old enough to post on r/metaphysics or something like that. Curious what you guys think.

hi well if you care i have a video of me solving 1/0 using a new variable i created V "Being at the edge of reality theorizing and inventing even when no one cares"

More generally, when a result has not yet been published, what kind of Turing-machine-like algorithm could quantify the importance of a result in pure mathematics in a way comparable to how humans do so?

Is a good mathematical result a question together with an answer (where here, the answer is taken to be the one with the shortest description length among the answers satisfying the question)? If so, is the description length of that question smaller than that of every other question that has the same answer as its answer?

we are everything we‘re not. if x=x and x is what defines someone and makes them the person they are and y is what they dislike and hate, it‘s still something that defines this person
so x=x=y
for example, if i would dislike blueberrie, that would make me someone who hates blueberries, which would be part of me and my personality

for another example, imagine a picture of you in a certain surrounding like a forest. you are the main subject in the picture and the forest is in the background. you are x and the forest would be y. you are certainly not the forest but if there wouldn‘t be any forest or any background what would define the line between you and the background, there isn‘t a certain border. so the forest still defines you somehow.

i‘ve come to the conclusion that mathematics are just a metaphor for things others rather describe in words. variables are metaphors, as the human has always deeply felt a need to describe his environment.
and in the example of schroedinger‘s cat, it isn‘t really about the cat being dead or not, but rather about the paradoxon of life and x being y at the same time.

I’m interested in whether structuralism genuinely explains mathematical necessity, or whether it simply relocates the ontological question. If structures are abstract, what ultimately grounds their existence and the truth of the relations within them?

hi, I just want to point out that ive been working all year to find out that sacred geometry is the building structure of the universe, proven in every domain, and will be releasing a universal problem solver based on yours's stuff.

clueless scientists won't look at it.

https://zenodo.org/records/21876931
Is supplement to

Preprint:10.5281/zenodo.17740562(DOI)

Preprint:10.5281/zenodo.17564091(DOI)

Preprint:10.5281/zenodo.17802660(DOI)

Preprint:10.5281/zenodo.18040237(DOI)

Preprint:10.5281/zenodo.18040262(DOI)

Preprint:10.5281/zenodo.18063546(DOI)

posting here by suggestion of reddit, I don't care who gets it or not. this is bascally a warning to the institutions because the tool i'm about to release, nobody is ready for it and no one wants hear about it.

Hey everyone, ​I want to discuss the philosophical motivations behind each of the ZFC axioms. ​Axioms are mathematically true by definition, but what philosophical worldview actually justifies them? For example, does the Axiom of Infinity require strict Platonism, or is it just about our cognitive ability to imagine such concepts? What about the philosophical reasoning behind the Axiom of Choice or Regularity? ​I'd love to hear your thoughts on the reasoning that grounds these axioms, or get recommendations for philosophers who have deeply explored the "why" behind ZFC.

If we define integers as including the natural numbers and we define natural numbers as the number 1 and any other number obtained by adding 1 to it repeatedly. How do we define the number 1?

And a follow up question: Pretend that I come from an alternate universe where 1 doesn't exist and numbers are all defined as pairs of 2. So we have 0, 2, 4, 6... as the natural numbers in my universe. How could you explain to me what 1 is? Do Peano's axioms adequately define a unit?

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.)

Deep in the forest lived a fox who was widely known as a master chef. His kitchen always smelled of the most refined spicesand his dishes were considered true masterpieces of culinary art. But the fox had an ironclad principle - the absolute foundation of every single one of his meals was meat. With this ingredient, he conjured up the most incredible creations.

One day, a hare hopped past the fox's kitchen. He stopped, sniffed curiously, and observed the artfully arranged plates standing on the counter.

Dear Fox, said the hare, "your dishes look truly masterful and delicious. Tell me, can you also make me a nice, tasty salad?"

The fox smiled confidently, adjusted his Chefs hat, and nodded. "Yes, I certainly can. But I will, of course, need some kind of meat for that. What kind would you like as a base?"

The hare gently shook his head. 'But I don't eat meat at all. I would like something entirely without meat.'

The fox's eyes widened, and he stared at the hare in sheer disbelief. He put his kitchen knife aside and raised a paw instructively. "I am sorry, but that makes no sense! Without meat, you cannot make a juicy steak, age a delicious salami, or braise a perfect roast. I cannot prepare food without this wonderful meat, that is simply impossible. Just consider: without meat, we would not have all these magnificent and sublime dishes that I am able to prepare here every day!"

The Hare let his ears droop and slowly turned away. He was deeply disappointed, as he would have been very happy to eat something good without meat for once. The fox did not understand the problem. All these opulent dishes, the steak, the salami, and the roast, did not interest the hare at all. He did not even miss them. He would much rather have eaten other great things that manage entirely without this one ingredient.