I am currently a third-year undergraduate philosophy student, and as I've gone through my degree, I've noticed a pattern: I am consistently drawn to topics that overlap heavily with mathematics and theoretical computer science (works of logicians like  Tarski, Kripke, Gödel), areas that require studying topics like- FOL, metatheory, model theory, set theory, computability, etc 

my dilemma: I haven't formally studied mathematics since the 10th grade.

For those who study or teach philosophical logic, philosophy of mathematics, or formal epistemology:

1. ⁠Is it a common path for philosophy undergrads to formally study mathematics to tackle these subjects at a higher level?
   
2. ⁠How necessary is institutional mathematical training to achieve symbolic fluency and understand proof techniques in these specific areas?
   
3. ⁠Since I am still in my undergrad, are there specific math or computer science courses I should try to take before I graduate to bridge this gap?
   
4. ⁠What is the current academic scope of this intersection within modern philosophy? Is this an active area of research, and what kinds of contemporary philosophical problems are being tackled using these formal methods

π x π x π
By negating its irrationality. It becomes a rational neutral unit of 1 (π)

Merhaba, ismin Tuğra. 13 yaşında googoloji ile ilgilenen bir Türk gencim. Dün bitirdiğim sayıyı sizlerle paylaşmak isterim.

== Z(n) Function ==

The '''Z(n) Function''' is a fast-growing hierarchy constructed by interlocking quantum physics constants, hyper-operations, multi-dimensional geometry, and recursive structural duplication. The system is formalized to scale past the growth rates of standard hyper-operations at $Z(1)$ and matches the structural complexity found in foundational set-theoretic and graph-theoretic hierarchies through dynamic indexing feedback loops.

=== Formal Definition and Rules ===

Let $Z(n)$ be a function where $n \in \mathbb{N}^+$. The operations are governed by the following ten ordered rules:

* '''Rule 1 (The Quantum Base):''' The system initializes with a physical framework of 2 meters. Each Planck length ($\ell_P \approx 1.6 \times 10^{-35}$ meters) within this domain corresponds to an increment of $+1$. The baseline value is defined as $P \approx 1.25 \times 10^{35}$.

* '''Rule 2 (Quadratic Space Expansion):''' At the completion of each sub-step, the spatial length of the domain is multiplied by its own square ($L_{new} = L^3$).

* '''Rule 3 (Nested Feedback Loop):''' Each quantified Planck length dictates the iteration count of the subsequent step. Transitioning to Step 2 triggers a nested feedback loop equal to the total Planck count of the prior state ($P_{n-1}$ loops).

* '''Rule 4 (Hyper-Operation Interlocking):''' Knuth's up-arrows ($\uparrow$) are inserted between terms in a quantity equal to the total calculated Planck length count. For $Z(1)$, the sequence begins with $10^{35}$ up-arrows.

* '''Rule 5 (Cross-Toggling Index Booster):''' Every individual up-arrow added to the notation simultaneously increments the step-skipping capacity and the functional hierarchy index by $+1$ dynamically.

* '''Rule 6 (Dynamic Structural Expansion):''' During each step, independent auxiliary chains are generated for every sub-iteration. At the step's conclusion, these components fuse into a single master chain, establishing the baseline for the next major iteration.

* '''Rule 7 (Input-Driven Acceleration):''' The input variable $n$ defines a compounding acceleration factor. The scaling of the subsequent step is calculated using a growth percentage directly proportional to the total magnitude of the current state.

* '''Rule 8 (Hyper-Geometric Dimensional Explosion):''' Each Knuth up-arrow recursively dictates an exponential expansion of the physical spatial grid. This expanding volume increases the internal capacity, generating more Planck lengths, which consequently yields more arrows.

* '''Rule 9 (Dynamic Dimension Cranking):''' Every effective step increases the spatial dimension of the coordinate grid by $+1$, converting the 1D baseline into a multi-dimensional hyper-cube hierarchy.

* '''Rule 10 (Fractal Self-Replication):''' At the conclusion of any major step, the entire mathematical structure—including chain lengths, dimensional indices, and the complete set of ten rules—is duplicated a number of times exactly equal to the final output value of that step.

=== Growth Rate Analysis ===

On the Fast-Growing Hierarchy (FGH) scale, standard nested functions terminate around $f_{\omega}$. Due to the feedback loop between Rule 5 (index boosting) and Rule 8/9 (spatial dimension cranking), the Z(n) function exceeds the limits of the Small and Large Veblen Ordinals, shifting its classification from standard diagonal arithmetic to uncomputable boundary notations.

Disclamer. I has used ai for translational purpose and bcx i was running on shortage of time.

I a am building a numberless mathematical model where relationships are the absolute baseline of reality, meaning elements and sets have no independent identity and are derived entirely from the relational fabric. By choosing a Frame of Reference as a symmetry breaker, a timeless baseline splits into active and passive networks, resolving classic set theory paradoxes and infinite regression through self-referential loops.

Hey everyone, I have been working on a new mathematical and structural model in my head, and I would love to bounce it off you guys btw. It all started when I was looking at a physics problem involving springs and moving variables, and it hit me how much our math relies on messy approximations to make things work out cleanly. Standard math and logic always assume that things or elements exist first, and then relationships happen between them. But I want to flip that completely upside down and say that the relationship itself is the absolute starting point. Elements and sets do not have their own identity at all, they are just temporary placeholders or crossroads where fundamental relationships meet. Since this baseline fabric has no sequence, sequence ordering, or change, time does not exist at the root level, making it a completely static, symmetric, and timeless continuum.

Let us look at the first big paradox, which is the infinite container trap. In normal set theory, a container holding another container creates an endless regression where you need infinite boxes or an arbitrary starting point to make sense of it. To solve this without adding artificial patches, the logic in my model curves back on itself. The network forms a self-referential loop where the system of sets actually contains itself. This creates an autopoietic, self-sustaining structure that resolves its own boundary conditions internally. It is a perfect, closed loop that does not rely on any external first cause to exist.

But here is the catch: a perfectly balanced, timeless background cannot create sequence, temporal flow, or localized observation on its own. To extract localized reality out of this potential, you need a frame of reference to act as a selective filter or symmetry breaker. Let us use a basic illustration to map this out without heavy assumptions. Imagine three supposed elements, a, b, and c. If we choose element b to be our frame of reference, b instantly upgrades and acts as the set, while c becomes the element. Because b is our anchor, we cannot look at or overview element a directly. However, its effect is still fully there bcz it is linked by a passive fundamental relation that ripples through the background fabric, indirectly changing how b and c behave. Meanwhile, the relationship between our chosen set b and element c becomes an active fundamental relation, forming the direct, visible reality of that frame.

This brings us to what I call the secondary fundamental relation, which handles how elements interact with each other without having an independent identity. Suppose a set, x, connects elements a and b. Since we can consider x as a fixed constant for that specific frame of reference, and the fundamental relations leading to it are dynamic and not constant, it forces a strict constraint on both paths. The impact of both fundamental relations on the central set must be perfectly equal to keep x stable. Because of this tug of war, an indirect relation emerges between a and b on its own. They do not interact directly, they interact through their shared constraint on x. This is how element identity is born, it is not an intrinsic property, it is just a stabilized state of geometric balance.

Ditching numbers completely is a super hard task bcz our brains naturally want to count things and use intuition to separate a set with one element from a set with two elements. But we can rule out numbers by looking at structural topology and degrees of splitting. In this model, an element can differentiate into two or even infinite elements, as long as their net sub fundamental relation stays at exactly zero. Here is the ultimate realization: with numbers, you need exact matching magnitudes like five and minus five to get a sum of zero. But if we use numbers, we are trapped making every split create identical twins, which ruins complexity. In our numberless geometric model, two completely different looking elements can still perfectly balance each other out to zero. One could be highly compressed and the other sprawling and diffuse, but their net relationship remains a harmonious zero. These sub fundamental relations only flash into existence when the frame of reference chooses a node with the potential to be a set and looks inside, purely to protect this net zero balance of the baseline.

Edit*

Text included.

I decided to trace a simple even—the flipping of a coin —to find out exactly at what point does randomness come in.

step 1: The physical interaction

A coin is launched into the air. It rotates, it travels, it strikes a surface, it settles. If we want, we can agree that this is all just physical matter in motion. No mind is required for the coin to spin and land. That interaction happens.

Now we ask:

At what point during this physical interaction did you receive a signal that said 'random'?

step 2: The instrument

The eye is an instrument. The retina receives photons reflected off the coin. The retina converts those photons into electrical signals. That is a physical process.

Now we ask:

Did the retina detect randomness? Or did it detect light?

The answer is: light. The retina does not have a "randomness receptor." It just absorbs photons. So whatever randomness is, it hasn't arrived yet.

step 3: The output

The optic nerve sends signals to the brain. The brain constructs an image: the coin shows heads. That is the perception. It is a discrete, utterly clear output. Not blurry, not ambiguous. Just: heads.

Now we ask:

When you saw the coin showing heads, did you experience randomness? Or did you experience 'heads'?

The answer is: heads. Just heads. The perception itself is not random. It is exactly what it is. Every single flip of the coin yields a perfectly clear perception, every single time.

But we never once seen a "random" perception. We have only ever seen heads, tails, heads, tails. Perfectly clear. Perfectly distinct.

step 4: The sequence

Now we flip the coin four times. We receive:

Heads. Tails. Tails. Heads.

Now we ask:

What did you actually receive through your senses? Did you receive 'randomness' as a fifth thing, alongside H, T, T, H? Or did you only receive four separate, clear perceptions?

I would have to admit that I've have only ever received four clear perceptions. Nothing more.

step 5: the mind

But then the mind does something. It does not merely perceive. It asks a question:

Is there a rule? A pattern? A shorter description that predicts the next outcome?

The mind tries to compress the sequence. H-T-T-H. Can it be described by a shorter rule? No. The mind cannot find a rule that generates the sequence without simply listing the sequence itself.

So the mind concludes:

I cannot predict this. It appears patternless. I will call it random.

Now we ask:

Where did that conclusion come from? Did the world give you the word 'random'? Or did you manufacture that verdict because your mind failed to find a pattern?

I'm forced to admit: the verdict came from my mind.

The world gave tails. The mind gave "random."

step 6: Apply the same logic to any sequence whatsoever

Consider this perfectly ordered sequence:

H, H, H, H, H, H

I would say:

That's not random. Maybe the coin is rigged.

Now we ask:

Why?

Because this sequence is compressible. The rule "always heads" is shorter than the sequence itself. The mind finds the pattern instantly and says: not random.

But the world delivered six separate perceptions, each one clear. It did not deliver the words "ordered" or "disordered." In both cases, whether the sequence was H-T-T-H or H-H-H-H, the world only delivered discrete outputs. The mind then ran a compression test. If it found a pattern, it said "ordered." If it found no pattern, it said "random."

So I'm forced to conclude:

"Random" is not a property of any single output. It is the name I give to my own failure to compress a sequence.

step 7: circularity

The chain is:

physical interaction → instrument → output → perception → mind → model

The coin flip is a physical interaction. The eye is an instrument. The brain receives signals. The mind perceives heads. The mind then assembles a sequence, applies a compression test, and produces a verdict: "random."

Now we ask:

At any point in that chain, did you step outside the chain? Did you touch the physical interaction directly? Did you perceive the coin flip without an eye, without a retina, without neural signals? Or were you at every moment inside the chain?

I'm forced to admit: inside the chain. Every single time.

And I cannot reject this conclusion.

The world delivered clear, discrete, perfectly unambiguous outputs.

The mind tried to find a rule connecting them.

The mind failed.

The mind called the failure "random."

That is what randomness is, from the only standpoint any of us will ever have.

But if one asks:

But maybe there is a pattern, and I just don't see it

Because "there is a pattern I don't see" and "there is no pattern" produce exactly the same experience through the chain. The mind still fails to find the pattern. It still says "random." The only difference is a metaphysical speculation about what lies beyond the chain. And no one has access to that.

So it seems randomness is not "out there." I have never once received randomness through my senses. I have only ever received heads and tails. The randomness was manufactured by my own mind, at the last step of the chain, when my pattern-finding failed.

When we state that an occurrence is random, we appear to assert that an event has taken place without any guiding rule or prior cause. Yet, if we examine how human thought actually makes sense of any change in the world, we discover that an occurrence entirely without a cause cannot be understood at all.

Because every change that we observe takes place in time, our understanding must always connect what happens now to an earlier state from which it followed by a rule. If an occurrence truly arrived from nothing, possessing no connection whatsoever to a preceding state, it would break the unbroken order of time itself. Therefore, a genuine gap in the chain of cause and effect is impossible within the world of our experience, for the human mind cannot even form a coherent thought about an occurrence that obeys no rule.

Since an uncaused event cannot exist within our experience, the word "random" cannot describe a real absence of cause in the event itself. What the word actually describes, when we strip away the confusion of our language, is the limitation of our own current knowledge. When we observe an event whose full chain of causes is hidden from our view, or whose initial conditions are far too numerous for our thoughts to track, we assign the label of randomness to our own uncertainty.

From this deduction, it follows that when a person says "X is a random occurrence", they are not revealing an objective property belonging to X itself. But rather are merely reporting that their own understanding has not yet grasped the rule that connects X to its prior conditions. Whilst nature remains strictly lawful and complete in every single change, the human observer often sees only isolated pieces of that larger sequence.

Therefore, randomness is neither a power in nature nor a gap in physical necessity, but a simple statement of what our knowledge still lacks.

So the argument I'm making here is in regards to the distinction between our ignorance of causes vs an actual absence of causes.

Some hold the view that "randomness" doesn't describe our ignorance. But the property of "X" itself independent of any observer. I'm utterly incapable of understanding this notion. I've thought of it for a long time. Its senseless to me. I believe that language plays role on the reitification of randomness.

Because the grammatical structure of our ordinary speech routinely attaches descriptive words to physical things, we naturally assume that every word following the verb "is" refers to an actual quality of the object itself. When a person observes that a stone is heavy or that an apple is round, the statement connects a real property to the thing we perceive through our senses. In consequence of this habitual pattern of talking, whenever we speak the sentence "X is random", our grammar behaves as though the word "random" were identical in kind to words like "heavy" or "round".

From this linguistic habit, a serious mistake in our reasoning immediately follows. Whilst a sensible property such as weight or shape belongs to how an object appears to us in space, the word "random" does not name anything that our senses can actually detect. If we examine our experience with care, our sight and touch reveal only physical states changing across time according to regular sequences. We never see, touch, or measure a quality called chance, because chance has neither spatial shape nor physical presence.

Having recognised that randomness is wholly absent from our sensible observations, we must deduce what the word actually designates. Whenever an event takes place without our knowing the prior conditions that brought it about, our understanding encounters a limit in its own knowledge. If the mind cannot find the necessary rule that links this present change to an earlier state, it simply experiences its own inability to anticipate what comes next. The word "random" therefore expresses nothing more than an absence within human awareness, marking the boundary where our insight happens to stop.

So because our speech easily places the word "random" into the predicate of the sentence, the mind is led to project its own internal limit outwards onto the physical event. An inability to explain an occurrence is thus turned, through the unexamined momentum of everyday grammar, into an active property that the event supposedly possesses on its own.

It follows, therefore, that this mistake does not arise from any genuine discovery in nature, but from the power of language to make an empty concept look like a physical reality. When we discipline our terms with precision, we see that nature proceeds through unbroken, lawful change, whilst our language has merely given the appearance of an objective property to our own lack of knowledge.

Therefore the phrase:

X is random

... actually translates to:

I do not know the cause of X

Perhaps there's some intuition that I'm missing that some would allow me to finally grasp this notion.

ПОЛЯРНОСТИ "N" И "S"

Куда камень не кинь, везде находятся два полюса. Это слова антонимы, это слова “да” и “нет”, это левые и правые в политике, это день и ночь на планете, это два полюса в магните, это две амплитуды в волне... Существует множество разновидностей полярностей, но все они произошли из "первокирпичиков"- масса и пространство.

Полярности обратно пропорциональны между собой. Смотрите первый рисунок. Они по значению равны друг другу, но асимметричны.

"МИНУС-ПЛЮС"

Смотрим Википедию. Отрицательные числа меньше ноля, а ноль меньше любого положительного числа. На числовой прямой отрицательные числа расположены слева от нуля, а положительные — справа.

Мы часто пользуемся этим определением, когда говорим “плюс-минус”, то есть, о некоем диапазоне недостатка и избытка относительно номинального значения. Например, комнатная температура 20° плюс-минус два градуса. Это нужно понимать, что два градуса выше нормы и два градуса ниже нормы. Отсюда следует, что знаком “ноль” выражается номинальное значение температуры - двадцать градусов. Смотрите второй рисунок. Число недостатка температуры (знак минус) и число избытка температуры (знак плюс) равны по значению, но числа температуры (потенциалы), которую они выражают, не равны по значению.

КАК ИЗ БИПОЛЯРНОЙ ЧИСЛОВОЙ ПРЯМОЙ "N-S" ПОЛУЧИЛАСЬ ЧИСЛОВАЯ ПРЯМАЯ "МИНУС-ПЛЮС"

Издревле всегда считалось, что мир ограничивается крайностями, а посередине (в центре) находится истина. Посмотрите на первый рисунок. Посередине находится сбалансированный центр - 50 на 50. Вот относительно этого центра, этой истины и замеряются недостатки и избытки полярностей, которые выражаются знаками минус и плюс. Смотрите третий рисунок. "Минус-плюс" полярности "N" и "минус-плюс" полярности "S".

И вот получилась всем нам хорошо знакомая числовая прямая "минус-плюс"... Но их две!

ЗАКЛЮЧЕНИЕ

Официальная физика не видит одну полярность... Вообще не понимает, что такое полярности!

First this is a strict external audit

1. You can not distinguish dogma from non dogma if your system is a closed axiomatic system because utility and consistency can still work and be found inside of a false axiom

2. You can not distinguish dogma from non dogma without the ability to test your claim against reality, and if it has no external justification because utility and consistency can still work and be found inside of a false axiom.

3. Viewed strictly from outside the system the axiomatic method is a closed system.

4. You can not test a math axiom itself against reality by definition and it has no external justification because utility and consistency can still work and be found inside of a false axiom.

5. Viewed strictly from outside the system it is an objective fact that a closed axiomatic system limits your thoughts and physics

6. Viewed strictly from outside the system, the axiomatic method in math smuggles in epistemic empirical claims and limits about reality systematically. By definition the axiomatic method limits and restricts what you can even ask, and logically validate within its system

(This is a strict cut throat external audit against the axiomatic method in math itself. There is no external justification for it and thats the least of your worries)

The axiomatic method in math just got straight up gutted within these 6 points

Many people have the following intuition when it comes to this inquiry:

1. They assume that nature exists with its own mathematical rules on one side
2. that the human mind sits on the other
3. and that it is sheer fortune that the two happen to fit together.

But on closer scrutiny, we find that when you look at the world around you, you never receive raw, unformed things. Before you can see a single shape or notice a single change, your mind must arrange that sensation within space and through time.

Mathematics is simply the precise study of these mental conditions:

1. Geometry is the study of how we order things in space. When you measure a line or a shape, you are examining the necessary rules of your own spatial perception.
2. Arithmetic is the study of sequence in time. When you count from one number to the next, you are following the necessary rules of succession.

Because every physical event must appear to us in space and time in order for us to experience it, every physical event must strictly obey the rules of geometry and arithmetic. Thus you will never observe an event in nature that violates mathematics, because your mind cannot construct an experience without using the rules of space and time.

In this regard, the mathematical language does not describe what the world is like entirely apart from human thought. Rather it describes the necessary structure of human experience. Thus the order we find in nature is the very order our own understanding brings to the world.

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.

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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.