hopefully what you’re saying will be read by computer scientists, and mathematicians alike. I have found it interesting that so many people in computer science don’t see the connection between their neural networks and the brain and often warn people against making too much of that similarity. When indeed it is the opposite, we’re not using that similarity enough to understand our mind. Perhaps they’re just afraid to reduce our minds to that kind of mechanism. I think one mistake that so many people make when looking at this analogy as they forget that evolution has provided a pre-trained existing model in our brains. This 3 million years of pre-training has brought about the weights and connections in our brain that gives us a huge starting point compared to training a large language model from scratch. So when you think about the human mind, you can’t think that it operates with just the data that a child receives and compare that to the amountof data training atypical LLM, but rather the data that the species has been receiving for millions of years that went into building the current structure as well as the data streaming through the eyes of the newborn sensory system. I always look forward to your next post.
> so many people in computer science don’t see the connection between their neural networks and the brain and often warn people against making too much of that similarity. When indeed it is the opposite, we’re not using that similarity enough to understand our mind.
For every piece of advice (e.g. "Don't make too much of the similarity between neural nets and human brains"), there exists people for whom it is more appropriate to adopt the opposite advice (e.g. "We're not using that similarity enough to understand our mind.")
I guess just statistically/empirically, computer scientists are more likely to receive questions from people who are trying to understand neural nets than from people who are trying to understand human brains. And I guess statistically/empirically, in this situation people tend to rely on their intuitions or previous knowledge or metaphors about human brains, and apply it to try to understand neural net. And I guess if they're asking about this to computer scientists, it's because they're having trouble understanding something and asking for clarifications.
Given all this selection-effect, this is why computer scientists probably err on the side of the "Don't make too much of the similarity between neural nets and human brains" advice. I don't think it's because computer scientists "don’t see the connection".
I remember as a small child learning that our base 10 numbering system was not universal; that it was, in fact, completely arbitrary. This blew my mind and changed how I viewed mathematics. I enjoyed it and was good at it as long as the “why” behind it was evident (taught).
Periodically, I would get “stuck” on the ideas behind math rules, such as the concept and existence of zero. (See the terrific episode of the TV series about Young Sheldon—who goes on to win the Nobel Prize in Physics—on this, where he tries to teach his “delayed” young neighbor about zero and both he and his physics professors discuss how in the end we just have to pretend that the concept exists for math to work.)
Later I deduced, from teachers lacking in both imagination and the ability to teach the value of mathematics, that advanced math wasn’t for me. I can’t be the only person who was led, if you will, into functional innumeracy, in spite of loving Hofstader’s Gödel, Escher, Bach.
Your post here was helpful to me in understanding where and how I got stuck. I only wish I had read this at an early age. Instead, what I heard was, “Just trust me that this is true and real
and be sure to follow the rules strictly.”
That didn’t help once I realized that even our numbering system was arbitrary and made up. Also, there was a clear message that advanced math was a superior path to *all* others, and not really one for girls. Has this message changed much these days?
The parallel in reading and writing is that the arbitrariness is self-evident as soon as you are aware of other alphabets and languages. That, I could accept readily and see the reason and utility of. You are spot on that leavening the rules of math with some Marketing 101 would promote broader understanding. Thank you for being that promoter.
Count me as another person whose encounter with math in childhood leads me to nod vigorously at Bessis’ statement, “As an imaginary activity, math is a driver of neuroplasticity.” Numbering systems beyond Base 10 were just the gateway drug. Before I even graduated from high school, I was delving into non-Euclidean geometry and working through theorems to compare groups and loops.
For various reasons, including a boring summer job doing lab work, I pursued other career paths (broadcasting, desktop software), but I am still fascinated by whatever bits of math I encounter. As a spiritual teacher very interested in the formation of neural pathways, I often remind my community that the imaginal world is far larger than the physical world because it contains not only every past, present, and future thought about each aspect of the material world but also all thoughts about non-physical concepts — and thoughts about what we have not yet even imagined.
Surely you must be joking that you have to be able to define a subject in order to teach it! Only a straw formalist could say such a thing—tell my one subject that has such a definition and I’ll resign my job immediately.
What about this: "Astronomy is a natural science that studies celestial objects and the phenomena that occur in the cosmos."
If someone doesn't get astronomy, you can point at the stars.
If someone doesn't get math, what do you point at? Cryptic equations with funky Greek letters? Could it be the reason why so many kids mistakenly believe that math is the study of cryptic equations with funky Greek letters?
Regarding your job search, feel free to use the comment section. I'm sure that readers will offer some help.
What’s wrong with just teaching the question? Letting the students’ curiosity and imagination churn as the brain flexes and forms? Looking back on my long ago youth, I could “do” math to get a grade or pass a test but as I progressed, the certainty surrounding it really put me off. The question is always an essential component of the “answer.”
I don’t really understand the apparent claim that understand what the content of math is a blocker for younger students, either, though it’s more plausible for students at and a couple years beyond the calculus level—is that the level you’re interested in here? For the main mass of students, characteristic example is that many students in middle school, high school, and early college can’t handle fractions. Perhaps they’re not doing a good enough job of imagining combining 2/7 of a pie with 1/3 of a pie when adding, but I frankly doubt that’s relevant; I’d argue that conceptual understanding of foundational algorithms is built on top of practical facility with those algorithms moreso than the other way around. And anyway, I’m not sure how much students are losing energy mooning around about whether there really is a 2:7 floating out there in the Platonic realm.
So you think that a major part of the problem with math education is that teachers are preoccupied with Platonism vs formalism? If I’ve read you right, I continue to vigorously disagree.
I don’t see any qualitative sense in which that’s a more satisfactory definition than something like “math is the formal study of shape and quantity” or any number of other evocative though partial partial definitions of mathematics. (Is general relativity part of astronomy? What about simulations of cosmoses that may or may not align with our own?) Your post’s point seems to be much more about the importance of imagination to the practice of mathematics than about anything to do with giving a final, satisfactory definition of mathematics, which seems to be a fool’s errand.
Look, if this was a satisfactory definition of math, there wouldn't be caveats in dictionaries and encyclopediae. You never find such remarks under astronomy.
Amusingly, that “no consensus” sentence is not in the English Wiki article on mathematics as I look now. So it seems there’s no consensus on whether there’s consensus! I don’t see anything like that in the first couple of dictionaries I check, either. Anyway I most certainly don’t claim it’s a satisfactory definition, just that it’s no less satisfactory than your suggestion for astronomy.
Sorry, but that's just too much bad faith for me, or too much laziness and stupidity.
Don't you know that modern computers allow you to search in a page? The sentence has simply been moved to a dedicated section in the article.
I'm quoting the passage below for your convenience, but this is my last reply to you. Go trolling someone else.
"There is no general consensus about the definition of mathematics or its epistemological status—that is, its place inside knowledge. A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable. There is not even consensus on whether mathematics is an art or a science."
This is brilliant David, thank you for summarising it so neatly.
The difficult part for me is always knowing which mathematics tools are needed to describe the concepts (especially when the concepts that need describing are incredibly complex and self-interacting, as in quantum gravity) — in this sense, learning new maths is like learning new vocabulary, and (just as with language) the more advanced the concept, the more technical the vocabulary you must learn.
I don't think your perspective is that far removed from the mathematics as embedded a priori cognition "philosophy". Kant's arithmetic as the processing of time and geometry of space is one of his few cases of a priori reason. Personally I found "Where Mathematics Comes From" by Lakoff and Nunez to be very insightful in presenting mathematics as based on the thought exercises of analogies applied to fundamental cognitive structures.
A fun philosophy is of Badiou (e.g. Being and Event): that mathematics IS ontology. When one does math one thinks or communicates reality. This can be presented as credible relative to the cultural and context contingent signifiers-chasing-each-other language games.
I wouldn't say that mathematics is an "imaginary" activity. Beaming down to the planet is an imaginary activity -- an activity done only in people's imaginations. "Imaginative" would be a better word in my idiolect; tho' "imaginary" is better for the purpose of generating clicks.
I'm on the fence on this, probably because I'm accustomed to using "imaginary" for things that are no less "real" than other things (imaginary numbers vs real numbers).
"imaginative" is indeed more accurate from a certain perspective, which you rightfully document, but it's also misleading because it kind of assume some degree of creativity/talent that make the term more elitist than what I mean (my dictionary defines it as "having or showing creativity or inventiveness")
I don't perceive the same negative undertones as you do in "imaginary"—I view imagination as a noble activity.
I hesitating for a while after reading your comment, but I'm going to stick for now with "imaginary" (I might change my mind in the future)
At least it provides a framework for explaining it, which still needs to be empirically validated: as conjectured in the piece, the effectiveness of playing the game of truth is likely to have a machine learning interpretation.
I remember reading on the neural inner workings of mathematicians. It turned out that in Math you use the same neural pathways that are used in social environments. Mathematicians 'see' math concepts as actual agents that converse, link up with and conflict with other math concepts, like people that talk and interact with eachother. So in essence, to mathematicians reading formulas is like watching the Bold and the Beautiful for regular people.
Interesting. I'm not a mathematician but I have a dilettante interest in the problem of universals, and I saw this linked on HN.
ISTM that fundamentally, conceptualism suffers from the same problem as nominalism, which is that it implies your thoughts don't refer to anything outside your mind. Therefore any statement you make, any predication whatsoever, can only refer to what's in your mind. This includes mathematics. The statements "2+2=4" and "2+2=5" would simply refer to thoughts in the speaker's own mind, and go no further. And therefore, nothing is right nor wrong, and any idea of math as a body of knowledge rather than a mere description of mental processes (like emoting or sensing pain) is gone.
Yet we know this isn't so, because we observe mathematical truth at every level of reality. We teach 3 year olds how to count using extra-mental objects like apples. We build bridges using mathematical concepts. Etc. We observe the extra-mental reality of mathematics almost constantly, and so it must be taken as a datum, and any theory that contradicts it must therefore be rejected. Saying that math does certain things to our brain, which it undoubtedly does, does not change this.
I notice you don't distinguish between Plato's realism, and Aristotle's (probably to save space). The latter holds that universals *only* extramentally inhere in the objects which instantiate them. So as far as math goes, quantity is extramentally real, and the relations between different quantities that math describes are real, but they are only real when predicated of something else; they don't exist in some entirely abstract realm. "Five" doesn't exist, but "five apples" do. Nonetheless, quantity is universal, since everything we sense is quantified in various ways. This seems to avoid many of the problems with Plato's realism.
It's only from a Platonist perspective that "conceptualism suffers from the same problem"... from a conceptualist perspective, it's not a problem, it'a feature.
What's interesting about math is that the constant cognitive effort to align these fallible human semantics with the external, rigid formalism leads to mental images with unique stability and consistency across humans.
So, in fact, it's precisely in a conceptualist framework that one can understand the need for mathematics.
>It's only from a Platonist perspective that "conceptualism suffers from the same problem"... from a conceptualist perspective, it's not a problem, it'a feature.
ISTM that it's a problem regardless of perspective, Platonist or otherwise. Math must be extra-mental. The most basic observation of reality tells us this. If math were merely something that goes on inside our minds, bridges wouldn't be built, people would die all the time because nobody would know (or ever be able to know) what dose of medicine to give, nobody would be able to predict an object's speed based on its starting speed and acceleration, and so on. They may be able to say something about these things, but they would only be talking about the concepts in their own minds. Yet clearly this isn't the case. Therefore, any theory that claims math is purely mental -- as conceptualism does -- must be false.
No, you're missing an essential part of the story.
My take isn't that "math is purely mental". It's that the semantic side of math is a cognitive phenomenon. The formal side of math is obviously extra-mental.
What conceptualism brings on top of formalism/nominalism is that the seemingly gratuitous rules of formal logic are beneficial because of their favorable effects on conceptual reorganization.
Thank you for sharing this and for your wonderful book! I wish I had these when I was learning math in school, as some sort of mental scaffolding to build on.
Nice. A very similar definition, it seems to me, to the nicely self-referential one posited by Eugene Wigner in his essay, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.”
“… I would say that mathematics is the science of skilled operations with concepts and rules invented just for this purpose.”
hopefully what you’re saying will be read by computer scientists, and mathematicians alike. I have found it interesting that so many people in computer science don’t see the connection between their neural networks and the brain and often warn people against making too much of that similarity. When indeed it is the opposite, we’re not using that similarity enough to understand our mind. Perhaps they’re just afraid to reduce our minds to that kind of mechanism. I think one mistake that so many people make when looking at this analogy as they forget that evolution has provided a pre-trained existing model in our brains. This 3 million years of pre-training has brought about the weights and connections in our brain that gives us a huge starting point compared to training a large language model from scratch. So when you think about the human mind, you can’t think that it operates with just the data that a child receives and compare that to the amountof data training atypical LLM, but rather the data that the species has been receiving for millions of years that went into building the current structure as well as the data streaming through the eyes of the newborn sensory system. I always look forward to your next post.
> so many people in computer science don’t see the connection between their neural networks and the brain and often warn people against making too much of that similarity. When indeed it is the opposite, we’re not using that similarity enough to understand our mind.
For every piece of advice (e.g. "Don't make too much of the similarity between neural nets and human brains"), there exists people for whom it is more appropriate to adopt the opposite advice (e.g. "We're not using that similarity enough to understand our mind.")
See https://slatestarcodex.com/2013/06/09/all-debates-are-bravery-debates/ and https://slatestarcodex.com/2014/03/24/should-you-reverse-any-advice-you-hear/ for a deeper exploration on that idea.
I guess just statistically/empirically, computer scientists are more likely to receive questions from people who are trying to understand neural nets than from people who are trying to understand human brains. And I guess statistically/empirically, in this situation people tend to rely on their intuitions or previous knowledge or metaphors about human brains, and apply it to try to understand neural net. And I guess if they're asking about this to computer scientists, it's because they're having trouble understanding something and asking for clarifications.
Given all this selection-effect, this is why computer scientists probably err on the side of the "Don't make too much of the similarity between neural nets and human brains" advice. I don't think it's because computer scientists "don’t see the connection".
Brilliantly said.
I remember as a small child learning that our base 10 numbering system was not universal; that it was, in fact, completely arbitrary. This blew my mind and changed how I viewed mathematics. I enjoyed it and was good at it as long as the “why” behind it was evident (taught).
Periodically, I would get “stuck” on the ideas behind math rules, such as the concept and existence of zero. (See the terrific episode of the TV series about Young Sheldon—who goes on to win the Nobel Prize in Physics—on this, where he tries to teach his “delayed” young neighbor about zero and both he and his physics professors discuss how in the end we just have to pretend that the concept exists for math to work.)
Later I deduced, from teachers lacking in both imagination and the ability to teach the value of mathematics, that advanced math wasn’t for me. I can’t be the only person who was led, if you will, into functional innumeracy, in spite of loving Hofstader’s Gödel, Escher, Bach.
Your post here was helpful to me in understanding where and how I got stuck. I only wish I had read this at an early age. Instead, what I heard was, “Just trust me that this is true and real
and be sure to follow the rules strictly.”
That didn’t help once I realized that even our numbering system was arbitrary and made up. Also, there was a clear message that advanced math was a superior path to *all* others, and not really one for girls. Has this message changed much these days?
The parallel in reading and writing is that the arbitrariness is self-evident as soon as you are aware of other alphabets and languages. That, I could accept readily and see the reason and utility of. You are spot on that leavening the rules of math with some Marketing 101 would promote broader understanding. Thank you for being that promoter.
Count me as another person whose encounter with math in childhood leads me to nod vigorously at Bessis’ statement, “As an imaginary activity, math is a driver of neuroplasticity.” Numbering systems beyond Base 10 were just the gateway drug. Before I even graduated from high school, I was delving into non-Euclidean geometry and working through theorems to compare groups and loops.
For various reasons, including a boring summer job doing lab work, I pursued other career paths (broadcasting, desktop software), but I am still fascinated by whatever bits of math I encounter. As a spiritual teacher very interested in the formation of neural pathways, I often remind my community that the imaginal world is far larger than the physical world because it contains not only every past, present, and future thought about each aspect of the material world but also all thoughts about non-physical concepts — and thoughts about what we have not yet even imagined.
I like living in a universe that big.
Thanks for amplifying my thoughts. As a septuagenarian, I still faint when I hear some one saying 'Maths'.
How about: Mathematics is the science of quantifying the universe and its phenomena in useful ways.
That's physics and engineering. ;-)
Surely you must be joking that you have to be able to define a subject in order to teach it! Only a straw formalist could say such a thing—tell my one subject that has such a definition and I’ll resign my job immediately.
What about this: "Astronomy is a natural science that studies celestial objects and the phenomena that occur in the cosmos."
If someone doesn't get astronomy, you can point at the stars.
If someone doesn't get math, what do you point at? Cryptic equations with funky Greek letters? Could it be the reason why so many kids mistakenly believe that math is the study of cryptic equations with funky Greek letters?
Regarding your job search, feel free to use the comment section. I'm sure that readers will offer some help.
What’s wrong with just teaching the question? Letting the students’ curiosity and imagination churn as the brain flexes and forms? Looking back on my long ago youth, I could “do” math to get a grade or pass a test but as I progressed, the certainty surrounding it really put me off. The question is always an essential component of the “answer.”
I am always having to keep from laughing...
'astrophysics and the chaotic universe' ...
clearly illustrated with 2D geometry and counting math!
right.
I don’t really understand the apparent claim that understand what the content of math is a blocker for younger students, either, though it’s more plausible for students at and a couple years beyond the calculus level—is that the level you’re interested in here? For the main mass of students, characteristic example is that many students in middle school, high school, and early college can’t handle fractions. Perhaps they’re not doing a good enough job of imagining combining 2/7 of a pie with 1/3 of a pie when adding, but I frankly doubt that’s relevant; I’d argue that conceptual understanding of foundational algorithms is built on top of practical facility with those algorithms moreso than the other way around. And anyway, I’m not sure how much students are losing energy mooning around about whether there really is a 2:7 floating out there in the Platonic realm.
I never said the issue was mostly with students not understanding what math is about. The issue is primarily with teachers not knowing it.
So you think that a major part of the problem with math education is that teachers are preoccupied with Platonism vs formalism? If I’ve read you right, I continue to vigorously disagree.
I don’t see any qualitative sense in which that’s a more satisfactory definition than something like “math is the formal study of shape and quantity” or any number of other evocative though partial partial definitions of mathematics. (Is general relativity part of astronomy? What about simulations of cosmoses that may or may not align with our own?) Your post’s point seems to be much more about the importance of imagination to the practice of mathematics than about anything to do with giving a final, satisfactory definition of mathematics, which seems to be a fool’s errand.
Look, if this was a satisfactory definition of math, there wouldn't be caveats in dictionaries and encyclopediae. You never find such remarks under astronomy.
Amusingly, that “no consensus” sentence is not in the English Wiki article on mathematics as I look now. So it seems there’s no consensus on whether there’s consensus! I don’t see anything like that in the first couple of dictionaries I check, either. Anyway I most certainly don’t claim it’s a satisfactory definition, just that it’s no less satisfactory than your suggestion for astronomy.
Sorry, but that's just too much bad faith for me, or too much laziness and stupidity.
Don't you know that modern computers allow you to search in a page? The sentence has simply been moved to a dedicated section in the article.
I'm quoting the passage below for your convenience, but this is my last reply to you. Go trolling someone else.
"There is no general consensus about the definition of mathematics or its epistemological status—that is, its place inside knowledge. A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable. There is not even consensus on whether mathematics is an art or a science."
Wow, touchy!
Had you considered call math an "abstract activity"?
Although not as provocative as imaginary, abstract expresses the idea that math drops some details of reality to convey an aspect of reality.
This is brilliant David, thank you for summarising it so neatly.
The difficult part for me is always knowing which mathematics tools are needed to describe the concepts (especially when the concepts that need describing are incredibly complex and self-interacting, as in quantum gravity) — in this sense, learning new maths is like learning new vocabulary, and (just as with language) the more advanced the concept, the more technical the vocabulary you must learn.
Thanks! I have been saying this since I was 4 years old, and instead of listening to me I was punished in the class. Always.
I don't think your perspective is that far removed from the mathematics as embedded a priori cognition "philosophy". Kant's arithmetic as the processing of time and geometry of space is one of his few cases of a priori reason. Personally I found "Where Mathematics Comes From" by Lakoff and Nunez to be very insightful in presenting mathematics as based on the thought exercises of analogies applied to fundamental cognitive structures.
A fun philosophy is of Badiou (e.g. Being and Event): that mathematics IS ontology. When one does math one thinks or communicates reality. This can be presented as credible relative to the cultural and context contingent signifiers-chasing-each-other language games.
It's rational pattern-finding.
I wouldn't say that mathematics is an "imaginary" activity. Beaming down to the planet is an imaginary activity -- an activity done only in people's imaginations. "Imaginative" would be a better word in my idiolect; tho' "imaginary" is better for the purpose of generating clicks.
Thanks for your comment!
I'm on the fence on this, probably because I'm accustomed to using "imaginary" for things that are no less "real" than other things (imaginary numbers vs real numbers).
"imaginative" is indeed more accurate from a certain perspective, which you rightfully document, but it's also misleading because it kind of assume some degree of creativity/talent that make the term more elitist than what I mean (my dictionary defines it as "having or showing creativity or inventiveness")
I don't perceive the same negative undertones as you do in "imaginary"—I view imagination as a noble activity.
I hesitating for a while after reading your comment, but I'm going to stick for now with "imaginary" (I might change my mind in the future)
Paul Graham said that the best definition of mathematics that he could think of was "Mathematics is the study of precisely defined terms"...
Does this view of mathematical activity explain better than the competitors why math is so useful in empirical sciences?
At least it provides a framework for explaining it, which still needs to be empirically validated: as conjectured in the piece, the effectiveness of playing the game of truth is likely to have a machine learning interpretation.
I remember reading on the neural inner workings of mathematicians. It turned out that in Math you use the same neural pathways that are used in social environments. Mathematicians 'see' math concepts as actual agents that converse, link up with and conflict with other math concepts, like people that talk and interact with eachother. So in essence, to mathematicians reading formulas is like watching the Bold and the Beautiful for regular people.
Interesting. I'm not a mathematician but I have a dilettante interest in the problem of universals, and I saw this linked on HN.
ISTM that fundamentally, conceptualism suffers from the same problem as nominalism, which is that it implies your thoughts don't refer to anything outside your mind. Therefore any statement you make, any predication whatsoever, can only refer to what's in your mind. This includes mathematics. The statements "2+2=4" and "2+2=5" would simply refer to thoughts in the speaker's own mind, and go no further. And therefore, nothing is right nor wrong, and any idea of math as a body of knowledge rather than a mere description of mental processes (like emoting or sensing pain) is gone.
Yet we know this isn't so, because we observe mathematical truth at every level of reality. We teach 3 year olds how to count using extra-mental objects like apples. We build bridges using mathematical concepts. Etc. We observe the extra-mental reality of mathematics almost constantly, and so it must be taken as a datum, and any theory that contradicts it must therefore be rejected. Saying that math does certain things to our brain, which it undoubtedly does, does not change this.
I notice you don't distinguish between Plato's realism, and Aristotle's (probably to save space). The latter holds that universals *only* extramentally inhere in the objects which instantiate them. So as far as math goes, quantity is extramentally real, and the relations between different quantities that math describes are real, but they are only real when predicated of something else; they don't exist in some entirely abstract realm. "Five" doesn't exist, but "five apples" do. Nonetheless, quantity is universal, since everything we sense is quantified in various ways. This seems to avoid many of the problems with Plato's realism.
Thanks for your comment.
It's only from a Platonist perspective that "conceptualism suffers from the same problem"... from a conceptualist perspective, it's not a problem, it'a feature.
What's interesting about math is that the constant cognitive effort to align these fallible human semantics with the external, rigid formalism leads to mental images with unique stability and consistency across humans.
So, in fact, it's precisely in a conceptualist framework that one can understand the need for mathematics.
Thanks for your response.
>It's only from a Platonist perspective that "conceptualism suffers from the same problem"... from a conceptualist perspective, it's not a problem, it'a feature.
ISTM that it's a problem regardless of perspective, Platonist or otherwise. Math must be extra-mental. The most basic observation of reality tells us this. If math were merely something that goes on inside our minds, bridges wouldn't be built, people would die all the time because nobody would know (or ever be able to know) what dose of medicine to give, nobody would be able to predict an object's speed based on its starting speed and acceleration, and so on. They may be able to say something about these things, but they would only be talking about the concepts in their own minds. Yet clearly this isn't the case. Therefore, any theory that claims math is purely mental -- as conceptualism does -- must be false.
No, you're missing an essential part of the story.
My take isn't that "math is purely mental". It's that the semantic side of math is a cognitive phenomenon. The formal side of math is obviously extra-mental.
What conceptualism brings on top of formalism/nominalism is that the seemingly gratuitous rules of formal logic are beneficial because of their favorable effects on conceptual reorganization.
Thank you for sharing this and for your wonderful book! I wish I had these when I was learning math in school, as some sort of mental scaffolding to build on.
Thank you Rachel!
Nice. A very similar definition, it seems to me, to the nicely self-referential one posited by Eugene Wigner in his essay, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.”
“… I would say that mathematics is the science of skilled operations with concepts and rules invented just for this purpose.”
https://www.maths.ed.ac.uk/~v1ranick/papers/wigner.pdf