i’m curious about the definition here of “thinking clearly” and being “intellectually capable”:
“What does it mean to be intellectually capable? There are many different ways of putting this. How many novel ideas have you put out there that have changed people’s thinking? How many discoveries have you made at the frontier of human knowledge? How many great essays/lectures have you produced that were viewed as insightful by other people who have produced insightful essays/lectures? How good are you at asking questions? How good at you at identifying ideas that are likely to be consequential in the future? How good are you at spotting intellectual grift?”
to me, these are all questions not of clarity or capability, but of influence and of thinking as a social activity. the questions here involve a sense of social others (either authorities, experts, or the general public), with capability defined as the ability to impact or influence these others using “thought”. it feels like you’re moving away from thought or truth as objective, verifiable, self-contained and self-justified (like pure mathematicians naively assume), and moving towards thinking as a collective, embedded, relational, social enterprise.
Why must math be rigid? I don't see dancing and stretching as contrary to mathematical thinking. As a math major in college, I would often read my problem set before bed so that my brain could start subconsciously working through the problems. Sleep was an essential step in my math problem-solving process.
Coincidentally, I once wrote a research report on Ramanujan. The whole divine inspiration from dreams thing may or may not be romanticized by Westerners and Indian biographers. His colleague Hardy saw him as a rationalist, and wrote that Ramanujan was “as sound an infidel as Bertrand Russell or Littlewood."
interesting point, and maybe this essay is really an example of a flawed conception of math. while I can appreciate the intuition aspect of math much more now, in college the thought of just reading the problem at night before sleeping would never have even crossed my mind!
do you still think math (and other fields that emphasize logic & precision like programming, analytical philosophy) demands rigidity more so than most other fields, despite it also requiring intuition? that's the point I still believe, and I wonder how much continuing to practice that precise/rigorous thought helps create clarity in other domains
Funnily enough, in college I often had the experience of *not* being able to solve the homework problem the day it was given, but being able to solve it the next day or the day after!
So it quickly became clear to me that "sleeping on it" totally worked.
Yeah, I can't really deny that math emphasizes rigidity relative to other fields.
I fear that letting go of rigor is dangerous in a way. Sometimes it is the only lifeline out of a confusing situation. Like if I am in a cult, getting confusing mixed signals, feeling the vibes and my intuition pointing me towards the group's beliefs. The only hope for getting out of the cult is demanding of yourself the highest standard of rigorous thinking. How else could you actually create clarity in such a scenario? I think the scenario is actually very common
totally. I do worry about letting go of rigor altogether; I think ideally there's an interplay between intuition and careful thought, and each can help inform/refine the other
> One misconception was the idea that math itself is a purer or more elevated kind of truth.
As far as I can tell mathematical truth is the only kind of truth that can be communicated to other people, because it is the only kind that can be independently verified. It seems important for that reason. (One could argue that dimensionless physical statements, like the total number of electrons in a neutral oxygen atom, can be independently verified too. But I think this is secretly a mathematical fact as well.)
> Does the fact that it’s more challenging lead you to actually get better at all other fields, at other kinds of thinking?
My personal experience is - yes. I've spent a lot of time learning a lot of different things and I frequently notice that math (and programming, for that matter) has vastly improved my ability to reason, and by extension, to write coherently. I've had periods in my life where I spent hours a day doing math and periods where I did none, and I have felt noticeably smarter in all aspects my life - especially in my ability to quickly grasp new concepts and situations - during the math-oriented periods.
I've also read many papers in origin of life research, and to be honest the quality of the thinking in those that attempt to propose theories without grounding in math is relatively poor. One side-effect of using mathematical reasoning is that it requires clear definitions of terms. In my experience many of these papers simply introduce poorly-defined terms ("edge of chaos" is a notable offender) and ascribe effects to those phenomena. This produces theories that are both unfalsifiable and make no predictions. In other words, they are useless. Math can be abused to produce useless theories as well (e.g. with spurious curve-fitting), but in those cases it's usually obvious.
> the fact that math is fundamentally about following logical rules, means you don’t get to do as much “flexing your intuition muscles”; thinking in math is a way of thinking very rigidly.
I think this is not true. Evaluating proofs is about following logical rules, and can be done by a computer. Coming up with proofs is about as intuitive an activity as any other (as you mention in the footnote).
> How do you make connections between disparate ideas? There is some kind of role here for relaxation.
Yeah, I agree.
> Math is a language of quantities devoid of qualities. It’s a language of form with no substance. It’s syntax without semantics.
If you take this position I think you also have to believe computers cannot be intelligent. Searle's Chinese Room argument.
> what role should math play, in a serious intellectual engagement with life?
My perspective is that math must play a fundamental role in such a life, for the reason above: that it seems to be the only way to communicate truth.
> how far will formal methods take us?
I guess the key here is whether the complexity of the human world is too great for formal methods - and I think the answer is probably no, not given sufficiently powerful computers and algorithms. Of course, there may be spiritual truths that remain inaccessible to math - math might not lead us to satisfaction and joy. But I'm not sure if that's what you're talking about.
> mathematical truth is the only kind of truth that can be communicated to other people
this seems needlessly limited to me, can't we communicate all kinds of truths to each other? if I say "I'm outside for a walk" is that not communicating truth?
> I have felt noticeably smarter in all aspects my life - especially in my ability to quickly grasp new concepts and situations - during the math-oriented periods.
intriguing, I haven't experienced this but I've never done more than 1-2hrs of pure math per day and/or I wasn't paying super close attention to how my cognitive abilities changed during that time
> Coming up with proofs is about as intuitive an activity as any other
true, and perhaps a bigger point of this essay should have been that math itself is not just about rigor and logic
> If you take this position I think you also have to believe computers cannot be intelligent
I don't think so, because I can just think of intelligence as a functional property (how you relate inputs to outputs, which indeed can be a purely syntactical property), whereas I think of consciousness as not purely functional – it depends somewhat on the substrate/implementation details
> there may be spiritual truths that remain inaccessible to math - math might not lead us to satisfaction and joy
maybe my thesis is: if the world is truly, deeply amenable to formal methods through and through, then even spiritual truths, satisfaction, joy etc should all be graspable with formal methods, given enough memory/computational power. if the universe itself is a mathematical object, then this would have to be true. but maybe memory/space constraints make this impossible to ever answer, I'm not sure if I'm even asking a real question at this point
“There is some kind of role here for relaxation. I’m not exactly sure why, but relaxing seems to allow for unexpected connections to be made”
I find that Iain McGilchrist speaks to this point beautifully in his “Matter With Things”. Essentially, what you call relaxation here, I map onto the right hemisphere, the ability to enter the state of broad, open awereness.
On another note, I’ve also been wanting to reclaim the idea of rigor for myself, and something I like is this idea of relational rigor, as an under-recognized twin of intellectual rigor. I tried to capture some of it here https://sandrasobanska.com/2022/07/29/case-for-relational-rigour/
relational rigor is very intriguing. I've been curious about McGilchrist's book for a while and just intimidated by its length, if you run into any good reviews/summaries let me know!
"I think the formalist point of view helps in explaining various aspects of mathematics.
What is maths? Maths is the study of symbol manipulation.
Why does maths work? Maths works because it focuses on symbol manipulations: extremely reliable and universal processes. Whenever we notice a process similar to some symbol manipulation, we can perform it ourselves (with a pen and a paper, for instance), and predict the outcome of that process.
When does maths not work? Maths fails when we fail to mimic a naturally occurring process occurring with symbol manipulation, or when we fail at understanding or correctly performing our own symbolic manipulation."
Interesting post! Lots I disagree with, though I'm biased as basically a mathematician :)
But it's mostly been well addressed by grant, Maia and Sid's comments.
>The way I could be wrong is if further study of pure math led to some groundbreaking discovery that had tangible implications for our understanding of the world.
Isn't special and general relativity a great example of exactly this? Einstein wouldn't have been able to come up with his theories without the differential geometry of the previous half century.
I'm planning to read Davis Bessis' book, will come back and comment more when I have!
> Isn't special and general relativity a great example of exactly this? Einstein wouldn't have been able to come up with his theories without the differential geometry of the previous half century.
yep, but that big delay between the math being discovered and subsequently being used, to me makes it clear that it wasn't *just* the study of math that led to GR, but the combination of pure math + a thoughtful, nontrivial application of it to an unexpected domain.
maybe it's a silly thing to ask for, but what I'm seeking in the original sentence you quoted, is a math proof that, just by virtue of being discovered, immediately leads to a breakthrough in science.
Paul Dirac created a relativistic quantum equation to describe electrons in 1928. His mathematical formulation unexpectedly produced negative energy solutions which seemed nonsensical at first. Rather than dismissing these as mathematical artifacts, Dirac interpreted them to predict the existence of "anti-electrons" (positrons) with the same mass but opposite charge. This purely mathematical prediction was confirmed experimentally when positrons were discovered by Carl Anderson in 1932, just four years later.
Black holes:
The Schwarzschild radius is the key mathematical insight that emerged from Karl Schwarzschild's solution to Einstein's field equations in 1916.
When Schwarzschild solved Einstein's equations for the gravitational field around a perfectly spherical mass, he found that at a specific radius (r = 2GM/c²), something mathematically strange happened:
The equations produced a mathematical singularity where certain terms seemed to become infinite
The time component and radial component of the metric tensor switched signs
This radius (now called the Schwarzschild radius) marks the event horizon of what we now call a black hole. The mathematics showed that:
Time dilation becomes infinite at this radius
The escape velocity equals the speed of light exactly at this radius
For any observer outside this radius, nothing that crosses this boundary can ever return
For a mass like our Sun, this radius would be about 3 kilometers. For Earth, it's just 9 millimeters.
The mathematical insight was that this boundary isn't just a peculiarity of the equations but represents a real physical boundary in spacetime where the properties of space and time fundamentally change. Later mathematical work by others showed this wasn't a coordinate artifact but a genuine feature of the spacetime geometry.
This mathematical boundary, derived purely from solving Einstein's equations, predicted the defining characteristic of black holes decades before any observational evidence existed.
Hawking radiation
The key mathematical insight came when Hawking calculated the spectrum of this radiation and found it precisely matched the spectrum of thermal radiation from a blackbody with a specific temperature directly related to the black hole's mass: T = ℏc³/(8πGMk).
This mathematical result immediately led to three profound scientific breakthroughs:
Black holes must have entropy and temperature, connecting gravity with thermodynamics
Black holes will eventually evaporate completely, contradicting the previous understanding
Information might be lost in black holes, challenging quantum mechanics' unitarity principle
I asked Claude, and it came up with Maxwell discovering light was an electromagnetic wave because it fell out of the equations:
Maxwell's critical mathematical insight was identifying and correcting a logical inconsistency in Ampère's Law. The original law stated that magnetic fields are produced by electric currents (∇×B = μ₀J). However, this equation violated conservation of charge when applied to capacitors.
Maxwell resolved this by adding a new term: the "displacement current" (μ₀ε₀∂E/∂t). This term represented the rate of change of the electric field over time.
What It Predicted:
When Maxwell incorporated this new term and combined all the equations, he could mathematically manipulate them to derive a wave equation:
∇²E = μ₀ε₀(∂²E/∂t²)
This equation described waves traveling at a specific speed: 1/√(μ₀ε₀)
When Maxwell calculated this speed using the known values of μ₀ (magnetic permeability) and ε₀ (electric permittivity), he got approximately 3×10⁸ meters per second—exactly matching the measured speed of light.
The Breakthrough:
This mathematical result directly revealed that:
Light is an electromagnetic wave
There should exist a whole spectrum of electromagnetic waves beyond visible light (radio waves, infrared, ultraviolet, etc.), all traveling at the same speed but with different wavelengths
These waves could be generated by oscillating electric charges
This wasn't extrapolation or interpretation—it fell directly out of the mathematics. The mere act of making the equations consistent revealed that electricity, magnetism, and light were manifestations of the same fundamental phenomenon.
i’m curious about the definition here of “thinking clearly” and being “intellectually capable”:
“What does it mean to be intellectually capable? There are many different ways of putting this. How many novel ideas have you put out there that have changed people’s thinking? How many discoveries have you made at the frontier of human knowledge? How many great essays/lectures have you produced that were viewed as insightful by other people who have produced insightful essays/lectures? How good are you at asking questions? How good at you at identifying ideas that are likely to be consequential in the future? How good are you at spotting intellectual grift?”
to me, these are all questions not of clarity or capability, but of influence and of thinking as a social activity. the questions here involve a sense of social others (either authorities, experts, or the general public), with capability defined as the ability to impact or influence these others using “thought”. it feels like you’re moving away from thought or truth as objective, verifiable, self-contained and self-justified (like pure mathematicians naively assume), and moving towards thinking as a collective, embedded, relational, social enterprise.
wow this sounds exactly right
Why must math be rigid? I don't see dancing and stretching as contrary to mathematical thinking. As a math major in college, I would often read my problem set before bed so that my brain could start subconsciously working through the problems. Sleep was an essential step in my math problem-solving process.
Coincidentally, I once wrote a research report on Ramanujan. The whole divine inspiration from dreams thing may or may not be romanticized by Westerners and Indian biographers. His colleague Hardy saw him as a rationalist, and wrote that Ramanujan was “as sound an infidel as Bertrand Russell or Littlewood."
What is math?
interesting point, and maybe this essay is really an example of a flawed conception of math. while I can appreciate the intuition aspect of math much more now, in college the thought of just reading the problem at night before sleeping would never have even crossed my mind!
do you still think math (and other fields that emphasize logic & precision like programming, analytical philosophy) demands rigidity more so than most other fields, despite it also requiring intuition? that's the point I still believe, and I wonder how much continuing to practice that precise/rigorous thought helps create clarity in other domains
Funnily enough, in college I often had the experience of *not* being able to solve the homework problem the day it was given, but being able to solve it the next day or the day after!
So it quickly became clear to me that "sleeping on it" totally worked.
Yeah, I can't really deny that math emphasizes rigidity relative to other fields.
I fear that letting go of rigor is dangerous in a way. Sometimes it is the only lifeline out of a confusing situation. Like if I am in a cult, getting confusing mixed signals, feeling the vibes and my intuition pointing me towards the group's beliefs. The only hope for getting out of the cult is demanding of yourself the highest standard of rigorous thinking. How else could you actually create clarity in such a scenario? I think the scenario is actually very common
totally. I do worry about letting go of rigor altogether; I think ideally there's an interplay between intuition and careful thought, and each can help inform/refine the other
> One misconception was the idea that math itself is a purer or more elevated kind of truth.
As far as I can tell mathematical truth is the only kind of truth that can be communicated to other people, because it is the only kind that can be independently verified. It seems important for that reason. (One could argue that dimensionless physical statements, like the total number of electrons in a neutral oxygen atom, can be independently verified too. But I think this is secretly a mathematical fact as well.)
> Does the fact that it’s more challenging lead you to actually get better at all other fields, at other kinds of thinking?
My personal experience is - yes. I've spent a lot of time learning a lot of different things and I frequently notice that math (and programming, for that matter) has vastly improved my ability to reason, and by extension, to write coherently. I've had periods in my life where I spent hours a day doing math and periods where I did none, and I have felt noticeably smarter in all aspects my life - especially in my ability to quickly grasp new concepts and situations - during the math-oriented periods.
I've also read many papers in origin of life research, and to be honest the quality of the thinking in those that attempt to propose theories without grounding in math is relatively poor. One side-effect of using mathematical reasoning is that it requires clear definitions of terms. In my experience many of these papers simply introduce poorly-defined terms ("edge of chaos" is a notable offender) and ascribe effects to those phenomena. This produces theories that are both unfalsifiable and make no predictions. In other words, they are useless. Math can be abused to produce useless theories as well (e.g. with spurious curve-fitting), but in those cases it's usually obvious.
> the fact that math is fundamentally about following logical rules, means you don’t get to do as much “flexing your intuition muscles”; thinking in math is a way of thinking very rigidly.
I think this is not true. Evaluating proofs is about following logical rules, and can be done by a computer. Coming up with proofs is about as intuitive an activity as any other (as you mention in the footnote).
> How do you make connections between disparate ideas? There is some kind of role here for relaxation.
Yeah, I agree.
> Math is a language of quantities devoid of qualities. It’s a language of form with no substance. It’s syntax without semantics.
If you take this position I think you also have to believe computers cannot be intelligent. Searle's Chinese Room argument.
> what role should math play, in a serious intellectual engagement with life?
My perspective is that math must play a fundamental role in such a life, for the reason above: that it seems to be the only way to communicate truth.
> how far will formal methods take us?
I guess the key here is whether the complexity of the human world is too great for formal methods - and I think the answer is probably no, not given sufficiently powerful computers and algorithms. Of course, there may be spiritual truths that remain inaccessible to math - math might not lead us to satisfaction and joy. But I'm not sure if that's what you're talking about.
very excited to discuss this more irl
> mathematical truth is the only kind of truth that can be communicated to other people
this seems needlessly limited to me, can't we communicate all kinds of truths to each other? if I say "I'm outside for a walk" is that not communicating truth?
> I have felt noticeably smarter in all aspects my life - especially in my ability to quickly grasp new concepts and situations - during the math-oriented periods.
intriguing, I haven't experienced this but I've never done more than 1-2hrs of pure math per day and/or I wasn't paying super close attention to how my cognitive abilities changed during that time
> Coming up with proofs is about as intuitive an activity as any other
true, and perhaps a bigger point of this essay should have been that math itself is not just about rigor and logic
> If you take this position I think you also have to believe computers cannot be intelligent
I don't think so, because I can just think of intelligence as a functional property (how you relate inputs to outputs, which indeed can be a purely syntactical property), whereas I think of consciousness as not purely functional – it depends somewhat on the substrate/implementation details
> there may be spiritual truths that remain inaccessible to math - math might not lead us to satisfaction and joy
maybe my thesis is: if the world is truly, deeply amenable to formal methods through and through, then even spiritual truths, satisfaction, joy etc should all be graspable with formal methods, given enough memory/computational power. if the universe itself is a mathematical object, then this would have to be true. but maybe memory/space constraints make this impossible to ever answer, I'm not sure if I'm even asking a real question at this point
“There is some kind of role here for relaxation. I’m not exactly sure why, but relaxing seems to allow for unexpected connections to be made”
I find that Iain McGilchrist speaks to this point beautifully in his “Matter With Things”. Essentially, what you call relaxation here, I map onto the right hemisphere, the ability to enter the state of broad, open awereness.
On another note, I’ve also been wanting to reclaim the idea of rigor for myself, and something I like is this idea of relational rigor, as an under-recognized twin of intellectual rigor. I tried to capture some of it here https://sandrasobanska.com/2022/07/29/case-for-relational-rigour/
relational rigor is very intriguing. I've been curious about McGilchrist's book for a while and just intimidated by its length, if you run into any good reviews/summaries let me know!
>I am less convinced that the way to understand the world is to make formal mathematical models of it and prove theorems about those models.
The best thing I have read on this and "unreasonable effectiveness" is https://cognition.cafe/p/formalism
"I think the formalist point of view helps in explaining various aspects of mathematics.
What is maths? Maths is the study of symbol manipulation.
Why does maths work? Maths works because it focuses on symbol manipulations: extremely reliable and universal processes. Whenever we notice a process similar to some symbol manipulation, we can perform it ourselves (with a pen and a paper, for instance), and predict the outcome of that process.
When does maths not work? Maths fails when we fail to mimic a naturally occurring process occurring with symbol manipulation, or when we fail at understanding or correctly performing our own symbolic manipulation."
thank you! I'll check out the formalism post, looks interesting
Interesting post! Lots I disagree with, though I'm biased as basically a mathematician :)
But it's mostly been well addressed by grant, Maia and Sid's comments.
>The way I could be wrong is if further study of pure math led to some groundbreaking discovery that had tangible implications for our understanding of the world.
Isn't special and general relativity a great example of exactly this? Einstein wouldn't have been able to come up with his theories without the differential geometry of the previous half century.
I'm planning to read Davis Bessis' book, will come back and comment more when I have!
> Isn't special and general relativity a great example of exactly this? Einstein wouldn't have been able to come up with his theories without the differential geometry of the previous half century.
yep, but that big delay between the math being discovered and subsequently being used, to me makes it clear that it wasn't *just* the study of math that led to GR, but the combination of pure math + a thoughtful, nontrivial application of it to an unexpected domain.
maybe it's a silly thing to ask for, but what I'm seeking in the original sentence you quoted, is a math proof that, just by virtue of being discovered, immediately leads to a breakthrough in science.
3 more examples:
Positrons/Antimatter
Paul Dirac created a relativistic quantum equation to describe electrons in 1928. His mathematical formulation unexpectedly produced negative energy solutions which seemed nonsensical at first. Rather than dismissing these as mathematical artifacts, Dirac interpreted them to predict the existence of "anti-electrons" (positrons) with the same mass but opposite charge. This purely mathematical prediction was confirmed experimentally when positrons were discovered by Carl Anderson in 1932, just four years later.
Black holes:
The Schwarzschild radius is the key mathematical insight that emerged from Karl Schwarzschild's solution to Einstein's field equations in 1916.
When Schwarzschild solved Einstein's equations for the gravitational field around a perfectly spherical mass, he found that at a specific radius (r = 2GM/c²), something mathematically strange happened:
The equations produced a mathematical singularity where certain terms seemed to become infinite
The time component and radial component of the metric tensor switched signs
This radius (now called the Schwarzschild radius) marks the event horizon of what we now call a black hole. The mathematics showed that:
Time dilation becomes infinite at this radius
The escape velocity equals the speed of light exactly at this radius
For any observer outside this radius, nothing that crosses this boundary can ever return
For a mass like our Sun, this radius would be about 3 kilometers. For Earth, it's just 9 millimeters.
The mathematical insight was that this boundary isn't just a peculiarity of the equations but represents a real physical boundary in spacetime where the properties of space and time fundamentally change. Later mathematical work by others showed this wasn't a coordinate artifact but a genuine feature of the spacetime geometry.
This mathematical boundary, derived purely from solving Einstein's equations, predicted the defining characteristic of black holes decades before any observational evidence existed.
Hawking radiation
The key mathematical insight came when Hawking calculated the spectrum of this radiation and found it precisely matched the spectrum of thermal radiation from a blackbody with a specific temperature directly related to the black hole's mass: T = ℏc³/(8πGMk).
This mathematical result immediately led to three profound scientific breakthroughs:
Black holes must have entropy and temperature, connecting gravity with thermodynamics
Black holes will eventually evaporate completely, contradicting the previous understanding
Information might be lost in black holes, challenging quantum mechanics' unitarity principle
I asked Claude, and it came up with Maxwell discovering light was an electromagnetic wave because it fell out of the equations:
Maxwell's critical mathematical insight was identifying and correcting a logical inconsistency in Ampère's Law. The original law stated that magnetic fields are produced by electric currents (∇×B = μ₀J). However, this equation violated conservation of charge when applied to capacitors.
Maxwell resolved this by adding a new term: the "displacement current" (μ₀ε₀∂E/∂t). This term represented the rate of change of the electric field over time.
What It Predicted:
When Maxwell incorporated this new term and combined all the equations, he could mathematically manipulate them to derive a wave equation:
∇²E = μ₀ε₀(∂²E/∂t²)
This equation described waves traveling at a specific speed: 1/√(μ₀ε₀)
When Maxwell calculated this speed using the known values of μ₀ (magnetic permeability) and ε₀ (electric permittivity), he got approximately 3×10⁸ meters per second—exactly matching the measured speed of light.
The Breakthrough:
This mathematical result directly revealed that:
Light is an electromagnetic wave
There should exist a whole spectrum of electromagnetic waves beyond visible light (radio waves, infrared, ultraviolet, etc.), all traveling at the same speed but with different wavelengths
These waves could be generated by oscillating electric charges
This wasn't extrapolation or interpretation—it fell directly out of the mathematics. The mere act of making the equations consistent revealed that electricity, magnetism, and light were manifestations of the same fundamental phenomenon.
https://open.substack.com/pub/clementpaulus/p/the-recursive-self?r=5c1ys6&utm_medium=ios