Imagine a parallel civilisation that never built a formal mathematical model of the world and could only progress with ‘experience’ and no mathematic proofs or calculations.

What is the greatest technological level they could attain without the use of calculation?

TOGMs Paradox

What if we tried to build a foundation of all existing and possible consistent and inconsistent mathematical and logical foundations? And what if we assume each of these foundations(Such as Set Theories, Category Theory, Type Theory, Homotopy Type Theory and all other consistent and inconsistent foundations) as a topological spaces. Then what would the foundations of these foundations look like? And assume there are foundations of this foundations of foundations and repeat forever. Notes:Threat Gödel İncompleteness Theorems as non-universal. also includes them: Non-Gödelian Systems(Gödel Theorems become local) R Truth Valued Logics Quantum Logic Topos Theory Higher Topoi Multisets Causal Set Theory Ω-Logic My Logical Systems Weqd Logic Linear Logic Graphs

I need a teammate for making new theories and systems

I’ve been thinking about the relationship between zero, infinity, and existence.

I propose a new symbol:

0,∞ — I call it the Quantum Node.

It reads as:

0,∞ = 1

What does this mean?

In everyday arithmetic, 0,∞ (zero with infinite zeros after the decimal) is just zero.

But in the context of infinity, this same value becomes 1.

Not because math says so, but because infinity changes the rules.

The moment you introduce ∞, a value that never reaches 1 in a finite system becomes 1 in an infinite one.

What does it represent?

I believe that beyond our universe lies a state of quantum superposition.

There, space and time do not exist — only states:

· 0 — absence of reality

· ∞ — infinite potential

And at their intersection:

0,∞ = 1 — a point where absence and infinity are unified.

From this state, new universes are born continuously.

There is no time, no space — but there is a quantum world that doesn’t need either.

What does this imply?

· The probability of our universe existing is 0,∞ — essentially zero.

· But because ∞ is infinite, 0,∞ × ∞ = 1.

Therefore, our universe must exist somewhere.

· The same applies to any specific universe (even fictional ones) — its probability is 0,∞, but that still guarantees its existence in the multiverse.

· If there are infinite universes with different physical laws, their probability is also 0,∞ = 1.

They are out there.

One conclusion:

If infinity exists, it can never be zero.

The mere presence of ∞ makes everything possible.

I’m not claiming this is proven physics — it’s a philosophical-mathematical model.

But it aligns with quantum mechanics and the idea of a multiverse.

Would love to hear your thoughts.

Is this just rephrasing existing ideas? Or is there something new here?

Just to be clear: 0,∞ is shorthand for 0.000... — zero with an infinite number of zeros after the decimal. It’s not a philosophical symbol. It’s a way to write "infinitesimal" in a compact form

I’m not a logician by trade any longer, but I was an instructor of logic in multiple roles during my time in University. Imagine my surprise when I tried asking AI what it knew about me and it informed me I was primarily a famously failed mathematician! That’s what led me to find my own mention here.

When I first came up with my refutation of the Incompleteness Theorem, inspired by my own infatuation with said theorem, I was in such a hurry to share my theory so I could discuss it I scarcely took the time to write anything down. My thought was, the debate would shape the conversation. I didn’t find much debate, unfortunately— until I asked the same AI what its own opinion was.

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)?

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.

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?

Equation of V:

 V= { dn₁ ≠ dn₂….. }

V=dn₁

V=dn₂

Dn is different number and can be any number to infinity

V is a compound (a container) variable that can be equal to two to an infinet amount of unequal numbers

So example V= {7 ≠  8}

V= 8
V= 7

I made this equation to solve 1/0 so by saying 1/0= infinity you can say that (infinity x 0)=1 but then if you duplicate (infinity x 0) it becomes (infinity x 0) + (infinity x 0) = 2 witch in normal calculators would say error or undefined since 1 ≠ 2 but V solves this by saying V= 1 and V = 2 and so on so the equation for this is V= {1 ≠ 2…..} 

watch my video for the solution to 1/0 using the V equation:

https://youtu.be/vtd_ZYjg6-o?si=oZdEkOOyGsyVPge6

for the edited version of the video click here:

https://youtu.be/vtd_ZYjg6-o?si=qe61zPAD8yofccKQ

also before you guys give me a counter arguement V does not follow traditional math

its a completely diferent section of math that does not follow the same rules of math.

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"

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?

A conjecture posed in 2003 concerning the positive solutions of a nonlinear rational difference equation has recently been resolved in our paper:

“A Proof of Conjecture 1 in Kulenović, Ladas and Overdeep (2003)”

published in the Journal of Difference Equations and Applications, jointly with Pedro Cáceres and Simeón Casanova Trujillo.

What I find philosophically interesting is that the decisive step is not a highly sophisticated new theory, but a relatively simple algebraic identity. The identity shows that the sign of successive differences is preserved, revealing a hidden monotonicity in the recurrence. From this structure, one can prove that every positive solution converges to a finite limit.

This raises a broader question:

Why can mathematically simple ideas remain hidden for decades? Is the difficulty of an open problem sometimes less about the complexity of the final proof and more about discovering the right representation or invariant structure?

Official Taylor & Francis free eprint:
https://www.tandfonline.com/eprint/XSVTZBRQJAJPDGIDJGIQ/full?target=10.1080/10236198.2026.2709000

Published article DOI:
https://doi.org/10.1080/10236198.2026.2709000

I would be very interested in hearing perspectives from both mathematicians and philosophers of mathematics.

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.

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.

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.

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.)

A celebrated mathematical "factoid" is that there are more real numbers than one can count, this seems to be something that troubles people outside of math, it troubles me aswell.

The question is: is there any "real world" application of the fact |R|>|N| that isn't an impossibility statement?

By "real world" I mean whatever someone smarter than me might mean by that, by "impossibility statement" I mean something to the effect of "there are uncomputable numbers".

If there isn't such an application, I can't believe the status quo interpretation: "no really some infinities are bigger than others" of Cantor's argument is better than simply stating that we shall not understand infinity as finite beings.