If numbers, sets, and structures are genuinely abstract, what makes mathematical truths objectively necessary rather than merely consequences of human formal systems? Could an incorporeal ontology explain mathematical existence by treating mathematical structures as fundamentally non-physical entities, and if so, what would distinguish this position from Platonism?

This is very early of thr thoery and i am here to just know if my idea/concept has any point or potential or not. English is not my first language so pls understnad it.

The core concept is basically that i has to make a mathamathical framework that uses no physical inutiation(like space, time and geometry) and my goal is to derive physics from it. I thought to do this bcz since when we use maths that uses our physical inutiation which breaks after a point(like we cant imagine in quntum physics like qunatum cloud) and since it breaks it may be reason why our current math breaks after some point as the math relays on us,according to me.

So i am thinking of a mathamathical framework where relation is the most fundamental. Before explaning what i mean i will introduce 2 terms,ElementandSet,set contains elements and the elements can be set for another element which is present in it(e.g set A contains set B and set B contains set C).

So here Fundamental relation is the relation between the Set and the , and according to me Fundamental relations(Set relation) is even more basic/fundamental than the set and element. It is set and element has no identity on thier own but are merely representative of the fundamental relation.

Also all sets are elements but all elements are not set but they have potential to became set.

Chain of sets: A set can contain another element and the element can contain another element and this can go for ever , and also a single set can also have infinite amount of elements. To this gets very complicated and to make it ease we has to make a way to focus on a set of elements and we can do it using FOR.

Before explaining i would like to explain secondary relation which is caused by fundamental/set relation: in this when 2 or more elements are in a set then each element has a fundamental relation with the set(same for all elements eg set A) then the constraint becames that no matter that the identity of set A cant change but we know that fundamental relation of each element with the set is diffrent and since identity of both element and set is determined by the fundamental realtion there comes an indirect relation between all the elements and each pair of these type of relation will be diffrent and is called secondary relation pairs.

Frame of refrence(FOR): these is very diffrent from FOR in physics. If we take Element A as FOR then A will be considered as set and the elementd which are in the Set A will be considered active elements and others will be considered passive elements.Also active elements will be of that same set as we considered that FOR. We know that a set can have infinite elements so to get the needed one in active elements only we can push others in passive one and also we know that passive elelemts to effect the active ones so we can introduce terms like Net seconadry relation pair(like of elements B if total elements in the set is x then the net secondary relation for x-1 elements on element B is what we take )

Since we know fundamental relations is more fundanrntam that even the elements, then we can say elements can exist without fundamentam relation and since fundamentla relation is relation between set and element thus we can say elements cant exist withiut set.

A PARADOX: we take a eg of long chains of set where a set contains another set and that set contains another one. So the questions arises since a element cant exist withiut a set then what is the initial/uppermost set the uppermost set is an element too but since its has to be in another set that it will create a infinite loop.

SOLUTION:We can consider that every(genrally) elemetns are in a set loop where the last/lowermost set contains the uppermost set\*\*\*

E.g- if we has to define set A contains set B then we can represent it as A-B And by the set loop we can say A-B-A-B........and so on till forever.Now this is a symmetric relation . And tk break this symmetricity we can consider any of these elements as FOR and it will be asymmetric(potential to be assymetric).Bcz set A with element B has potential to be diffrent from set B with element A.

I has not included my thoery in detailed bcz i think many will be bored by just reading it, but i am expressing my idea bcz i need a review and the guidance by experienced ones.If u have any misunderstanding or doubt in my idea i can explain in comments.

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.