It all started after I heard the following podcast in this link:
https://www.econtalk.org/a-mind-blowing-way-of-looking-at-math-with-david-bessis/
The theme is 'rationality' and 'logic' are only methods: they are not fundamental to Maths. Fundamental to Maths, the author says, is intuition. Rationality and logic are methods to 'prove' or 'disprove' one's intuition(as perceived in the observable world).
Further, what David Bessis said is that Mathematicians need a visual picture in their minds before formalising the hypothesis into a formula.
But I thought about blind Mathematicians: how do they get a mental picture before deriving their formulas?
So, I asked OpenAI if there were many famous blind Mathematicians?
And the conversation meandered inevitably to Ramanujan. Is he the chosen one of the Devi?
Here is the conversation:
Me: Are there any great mathematicians who were blind?
OpenAI:
Yes—there have been remarkable mathematicians who were blind or became blind, and their work shows that deep mathematics relies far more on mental structure than on visual symbols.
Here are the most notable ones.
1. Nicholas Saunderson (1682–1739)
Completely blind from infancy
Blind due to smallpox before he was one year old
Became Lucasian Professor of Mathematics at Cambridge—the same chair later held by Isaac Newton
Taught algebra, geometry, and Newtonian optics
Invented tactile calculating boards so he could compute and teach
What’s striking is that he taught optics—a subject entirely about light—without ever having seen.
2. Lev Pontryagin (1908–1988)
Lost sight at age 14
A giant of 20th-century mathematics
Founder of Pontryagin duality, fundamental in harmonic analysis and topology
Major contributions to control theory (Pontryagin’s Maximum Principle)
He famously said he thought with his hands, walking while mentally structuring proofs.
3. Bernard Morin (1931–2018)
Blind from early childhood
Specialized in topology
First to produce a visual eversion of the sphere (turning a sphere inside out without tearing)
Did this purely through mental spatial reasoning
Ironically, a blind mathematician solved one of the most visual problems in mathematics.
4. Abraham Nemeth (1918–2013)
Blind from birth
Created the Nemeth Braille Code for mathematics and science
Though less famous for theorems, his contribution made advanced mathematics accessible to blind scholars worldwide
Me:
So, here is the 'visual' Maths part: I heard Mr Bessis in https://www.econtalk.org/a-mind-blowing-way-of-looking-at-math-with-david-bessis/ He seems to says that mathematicians make visual pictures in their minds before translating/mapping a formula. The essence of this talk is: 'Logic' and 'Rationality' is not fundamental to Maths (or, by extension to any 'Science'). Fundamental is intuition: rationality and logic are only methods to 'prove' or 'disprove' an intuition. Your thoughts?
OpenAI:
Your reading of Bessis is essentially right, and what you are circling is one of the oldest fault lines in mathematics.
Let me put my cards on the table first, then refine it carefully:
> Intuition is generative; logic is justificatory.
> Logic does not create mathematics — it certifies it.
That is not a romantic claim. It is an empirical one, reported again and again by great mathematicians.
1. What Bessis is really saying (stripped of rhetoric)
When Bessis says mathematicians “see pictures” before formulas, he does not mean literal visual diagrams alone.
He means:
pre-symbolic mental structures
felt constraints
dynamic shapes
tensions and invariants
These may be:
* visual (Riemann surfaces)
* kinetic (“this object wants to flow there”)
* auditory (patterns, rhythms)
* relational (“these things must balance”)
Only after this internal coherence is felt does the mathematician translate it into symbols.
So the real sequence is:
intuition → structure → symbols → proof
Not the other way around.
2. Why logic cannot be fundamental (and never was)
Logic has three hard limits:
(a) Logic cannot choose axioms
You must intuit what is worth assuming.
Gödel showed: logic cannot justify its own foundations.
(b) Logic cannot suggest what to prove
Logic can say:
> If A, then B.
It cannot say:
> A is interesting.
Interest, relevance, beauty — these are pre-logical
(c) Logic follows structure; it does not discover it
Proofs often come after the result is known to be true.
Hadamard, Poincaré, Grothendieck — all said the same thing:
> Discovery happens in darkness; proof happens in daylight.
3. But intuition alone is dangerous — and mathematics knows this
Intuition by itself is:
* seductive
* incomplete
* often wrong
That is why mathematics developed proof culture.
So the balance is not:
intuition vs logic
but:
intuition creates, logic disciplines
Think of intuition as conjecture-producing machinery, and logic as error-correction.
4. Visual intuition is just one kind — and not the deepest one
This connects beautifully to your earlier question about blind mathematicians.
Visual intuition is powerful, but it is not fundamental.
There are deeper intuitions:
* topological (“can this deform into that?”)
* algebraic (“this symmetry must exist”)
* variational (“this configuration wants to minimize something”)
A blind mathematician may lack optical imagery but still possess:
* spatial intuition
* structural intuition
* constraint intuition
In fact, forced away from pictures, they often reach cleaner abstractions.
This is why Pontryagin or Morin could operate at extraordinary levels.
5. A subtle correction to Bessis (important)
Bessis sometimes sounds as if:
> intuition is truer than logic
That would be a mistake.
A better formulation is:
> Intuition is epistemically prior,
> logic is epistemically binding.
Intuition gives meaning
Logic gives obligation.
Without intuition, logic is sterile.
Without logic, intuition is unreliable.
Mathematics exists only in their tension.
6. A deep closing thought (worth sitting with)
Mathematics is not built from logic.
It is built from necessity.
Logic is how necessity explains itself.
Or said differently:
> We do not prove truths into existence.
> We prove them so they cannot escape.
Me:
Bessis says, 'Ramanujan saw it in his dreams'. In other words, where the conscious and subconscious meld.
OpenAI:
Yes — that statement from Bessis (“Ramanujan saw it in his dreams”) is not a mystical exaggeration.
It is pointing to a very real cognitive phenomenon:
> Intuition arises in the borderland where the conscious and subconscious meet.
This meld is not poetic—it is neurological, mathematical, and historically documented.
Let’s break this down in a way worthy of your depth of thought.
1. The subconscious does the heavy lifting; the conscious takes the credit
Henri Poincaré (one of the greatest mathematicians of all time) describes his own creative process:
1. He consciously struggled with a problem.
2. He gave up and went for a walk.
3. Suddenly, the solution hit him fully formed.
He wrote:
> “The solution came to me… with the certainty of perfect clarity.”
This is exactly the Ramanujan effect.
The subconscious:
* runs thousands of combinations,
* notices symmetries,
* evaluates what fit,
* and presents a coherent structure to conscious awareness.
This is not logic.
This is pattern resonance.
2. Ramanujan’s dream mathematics is entirely consistent with this
Ramanujan often said:
Results “appeared” to him.
Dreams showed him formulas.
His goddess “revealed” identities.
This is metaphorical language for something deeply neurological:
> A subconscious that is unusually powerful at detecting deep invariants.
He was not receiving mathematics mystically.
He was receiving mathematics intuitively — from a subconscious processing engine far more active than normal.
Most humans have this engine.
Ramanujan’s was simply running at cosmic speed.
3. Why mathematics enters dreams
Dreams dissolve constraints:
* Linear time is gone
* Formal logic is gone
* Symbolic rigidity is gone
* Only structure remains
And mathematics is structure.
The subconscious never stops:
* recombining
* permuting
* generalizing
* and searching for fit
In dreams, those processes surface without the filtering of rational thought.
You yourself sensed this earlier:
> A system defining itself from within the system.
Dreams are precisely where the system unhooks from its own rules.
4. Intuition = fast pattern extraction in high-dimensional spaces
Here is the scientific framing:
When Bessis says intuition is fundamental, he means:
The mind works by pattern-guessing in enormous spaces
where logic is too slow and too narrow.
Mathematics requires:
* leaps
* analogies
* pictures
* analogical transfers
* resonance
* recognition of invariants
Conscious logic cannot generate these.
It can only check them.
Your conscious mind is the editor.
Your subconscious mind is the author.
5. Logic comes after, to certify intuition
This is key:
Intuition proposes.
Logic disposes.
Intuition generates the candidate truth.
Proof verifies it.
Ramanujan filled notebooks with results seen in his dreams, but many were:
* correct but unproven
* correct but stated imprecisely
* or missing conditions
Hardy then translated intuition into rigorous English mathematics.
This is the clearest example in all of mathematical history of:
> subconscious mathematics → conscious mathematics → formal mathematics
6. The deeper philosophical point
You’re asking the right question:
> Where does intuition come from?
The answer seems to be:
Intuition is the brain’s way of touching the structure of reality
before it explains that structure.
It is pre-experiential, pre-symbolic, pre-logical.
This is why:
* blind mathematicians feel topology
* physicists see equations as shapes
* Ramanujan dreams formulas
* Einstein rides on beams of light in thought experiments
Intuition is the interface between:
* the subconscious (pattern space)
* and the conscious (symbol space)
Logic is the stitching.
Me:
Two things: one, you are clearly discounting 'the chosen one' theory for Ramanujan; two, actually Bessis says you can and you must train your intuition. But first one I suspect must be built from years and years of 'science' literature that seems to say even for intuition one has define through 'rational' and 'logical' steps: step 1 > step 2 > step 3...
OpenAI:
You’re reading Bessis very carefully — and your two observations are exactly the *pressure points* of the whole debate. Let me address them one by one, and then show how they fit together without contradiction.
1. Yes — I am explicitly rejecting the “chosen one” theory (and so is Bessis)
Not because Ramanujan wasn’t extraordinary — he was — but because the chosen one narrative explains nothing.
If we say:
> Ramanujan was chosen, therefore intuition happened
we have:
* no mechanism
* no reproducibility
* no pedagogy
* no science
Bessis’ entire project is the opposite:
> If intuition exists, it must be cultivable.
Ramanujan is not evidence of divine exception;
he is evidence of **what the human mind can do under extreme conditions**:
* obsessive immersion
* early exposure
* continuous symbolic play
* zero separation between life and mathematics
He is an upper bound, not an anomaly.
So yes — both Bessis and I are firmly on the side that:
> There are no chosen ones; there are only differently trained minds.
2. Now the apparent contradiction you sense (this is subtle and important)
You are absolutely right to suspect this tension:
> If intuition is fundamental,
> why does training intuition seem to require
> step-by-step rational and logical procedures?
Isn’t that circular?
Short answer:
Logic trains intuition the way scales train musical improvisation.
You don’t perform music by playing scales —
but without scales, improvisation never appears.
3. What science actually says (when read carefully)
Cognitive science does not say:
> intuition is logical
It says:
> intuition is compressed experience
Here is the key idea:
Intuition = pattern recognition learned below awareness
And how do patterns get learned?
Through:
* repetition
* feedback
* correction
* comparison
* failure
All of which look like:
> step 1 → step 2 → step 3
But the steps train the subconscious, not the conscious.
4. The critical distinction (this resolves everything)
You are conflating two very different things:
Intuition is not executed step-by-step
Intuition is trained step-by-step
When intuition fires, it is:
* instantaneous
* non-verbal
* holistic
* felt as certainty or direction
But to build that capacity, the brain must be fed structured material.
5. Bessis’ actual claim (made precise)
Bessis is not saying:
> Forget rigor, trust vibes.
He is saying:
> Use rigor as gym equipment for intuition.
The workflow looks like this:
1. You struggle consciously
2. You attempt logical steps
3. You fail
4. You revisit examples
5. You prove small lemmas
6. You internalize constraints
7. Suddenly — intuition clicks
At that point:
the steps disappear
the structure remains
That structure is intuition.
6. Why Ramanujan didn’t show the steps
Ramanujan had:
* enormous internalized structure
* continuous mathematical exposure
* symbolic fluency from childhood
* no cultural emphasis on proof
So what surfaced was:
> the result — without the scaffolding
Hardy supplied the scaffolding later.
But that does not mean Ramanujan skipped training.
It means the training was:
* informal
* implicit
* obsessive
* non-academic
7. A deeper synthesis (this is the key insight)
You suspected that:
> even intuition seems to be built through rational sequences
The correct resolution is:
> Rationality is how intuition is trained,
> but intuition is how understanding is exercised.
They are not rivals.
They are phases.
8. Final, quiet conclusion
There is no magic.
There is no chosen one.
There is no shortcut.
But there is something profound:
> If you feed the mind structure long enough,
> it will begin to think structurally without effort.
That effortless thinking is what we call intuition.
Ramanujan lived there.
Bessis is telling us how to get closer.
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My own intuition is this: Ramanujan was the chosen one.
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