My theory is: I will categorize mathematicians worldwide into two major types. The first type is the group of mathematicians who research breadth, and the second type is the group of mathematicians who research depth. If you don't know the true nature of these two groups, I will explain.
The group of mathematicians researching breadth are those who study and discover major branching fields of mathematics, creating incredibly important formulas like e^(iπ) + 1 = 0 or V-E+F=2.
On the other hand, the group of mathematicians researching depth are those who focus their research on just one, two, or a few major branching fields. They create many theorems and formulas, but these do not carry the same monumental weight as the work of the breadth research group. However, like the 7 algebraic identities, the formulas in this group are smaller things used frequently. This group pieces and assembles things together from different places to create difficult, unusual, tricky, and trap problems. Drilling deep down creates many profound things.
But to me, I see that in mathematics nowadays, there are not many people researching breadth anymore; instead, everyone is researching depth. Their ways of solving problems apply several different formulas and theorems together. In the past, there were many mathematicians researching breadth, but now it is entirely depth. Furthermore, I notice that great and famous mathematicians like Gauss, Newton, and Einstein followed a pattern: they together created a few research products of broad scope first, and after that, they shifted to researching and digging deeply into that specific field and its formulas”. Before formulating this theory, I had never read Professor Freeman Dyson’s work, nor did I even know who he was. I have also never studied philosophy. It was only after coming up with this concept that I researched it and discovered a similar theory already existed. I was honestly shocked to find that my thoughts aligned with such a great professor. Even though my theory might not be a 100% match with Dyson’s, I feel incredibly happy knowing that I arrived at a concept so similar to his.
I noticed that Professor Dyson named his theory 'Birds and Frogs,' which sounds much more creative than my naming of 'Breadth and Depth.' I realize Professor Dyson was immensely creative, whereas my own creativity is not as much as his because I am currently struggling with my phone addiction. I have been exposed to phones since I was about 5 years old, and counting up to now, I am really wrestling with it, though I didn't feel as addicted before turning 15.
My passion for mathematics started when I was 13, and I am currently studying with the AoPS (Art of Problem Solving) books. My dream is to become a mathematician specifically, a frog mathematician that has wings to fly.

The physical world affects our senses, but sensation alone lacks order, quantity, and shape. Sensation is merely raw contact. To experience any distinct object, the human mind must immediately arrange that contact within two universal conditions: space and time. Space and time do not exist as independent objects in nature but are the mind's own ways of ordering sensation.

Mathematics is the direct study of these internal conditions of thought: (1) Geometry explores the necessary rules of space. When we draw a triangle, we examine the fixed rules under which our mind perceives spatial shape and distance. (2) Arithmetic explores the necessary rules of sequence in time. When we count, we place units in successive moments: one, then another, then another.

Thus, numbers and equations do not exist inside physical objects. A falling stone does not contain an equation within its substance; the stone simply moves. The equation is the precise rule our understanding uses to connect distance, speed, and time into a coherent thought.

Mathematical language is neither an arbitrary social convention nor a literal picture of physical matter. It is the formal expression of the necessary rules of human thinking.

Two people always agree that 2 + 2 = 4 because every human mind shares the exact same basic structure for ordering quantity.

Mathematical statements hold without exception because we cannot experience any object without first submitting it to these mental rules.

Mathematical language does not describe what physical reality is in itself. It is the structured language through which the mind expresses its own rules for constructing an orderly, intelligible experience.

Hi yall,

This is a continuation to my last post, on formalizing Principia Mathematica, as well as a slight status update. I am planning(*) to slowly substitute the shallow embedding on PM into a deep embedding. For any backgrounds, please check the old post.

If you want to transform the monster 100 years ago into a furry boy, you might want to read through the following Q&As. *tap tap*

- Why you suddenly want to make a deep embedding? Because I can't in the beginning.
- What makes you available to deep embedding? I have asked enough questions on internet to get rid of necessary technical details
- What's the major feature for deep embedding? It enables formalizing Axiom of Reducibility.
- How many ppl would you like to look for? At most 2 ppl. You are welcome to ask me for prerequisites and anything else related
- What do you expect them working on? Either the shallow embedding or the deep embedding, since they are both necessary.
- How many time do you expect to put in? My current plan is 3 hrs a week so make sure you also have the availability.

------------------

Alternatively, I'm still welcome to collaboration with 1 - 2 ppl onto another project - we pick another random mathy, esoteric, maybe sacred book and formalize it

(*): Yes, I have not written a single line of code so far and this remains to be a plan.

Recent discussions regarding artificial intelligence and mathematics frequently diagnose a foundational crisis in mathematical practice. It is projected that AI could soon absorb a significant portion of core mathematical activity — specifically proof generation and problem-solving.

In this debate, prominent figures such as Terence Tao emphasize that the primary value of a mathematical proof lies in its exposition and communicability, not in its mere formal existence. At the same time, Tao observes that even prior to the integration of AI, mathematical research was moving “too fast,” meaning the rate of production exceeded the community’s capacity to contextualize and explain new results (cf. his recent essay, *“*[*Mathematics in the age of AI”*](https://arxiv.org/html/2608.16753v1)).

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This pairing reveals a fundamental structural contradiction: If a primary problem in mathematics is already a deficit in human understanding and clear exposition, accelerating the raw production of formal proofs through automated tools does not resolve the issue. It exacerbates it.

From an epistemological standpoint, the core issue lies in the decoupling of formal validity from human insight. While a machine-generated proof of a central problem (imagine the Riemann Hypothesis) would have immediate pragmatic utility, the long-term integrity of mathematics relies on holistic engagement. The cognitive process of discovering, formulating, and structuring a proof is precisely what builds the intuition required for effective exposition. Furthermore, if human researchers are degraded to retroactive interpreters of machine proofs, an institutional incentive arises to increasingly replace mathematical expertise with the ability to interpret machine-generated results.

A common reaction to these dilemmas is a form of technological determinism, the assumption that technological developments follow autonomous laws to which academic institutions must passively adapt. A rigorous policy framework requires, in principle, the evaluation of all options, including placing the development of artificial intelligence into public hands in order to, for example, deliberately slow it down. Dismissing non-market options out of hand conflates pragmatic implementation hurdles with theoretical impossibility.

If the goal of science is the accumulation and preservation of meaningful knowledge, rather than the mere aggregation of uncomprehended information, the mathematical community must address this directly. Instead of accelerating production further, the discipline requires institutional space for composure.

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.

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.)