I'm a little confused by what the philosophical stakes are here. I'm perfectly happy to acknowledge the role of imagination in mathematics, but I don't quite see how that refutes or contradicts Platonism.
Imagination is not, in general, producing just any old thing. It has constraints, even dreams and fictions have constraints. Mathematical imagination has sharper constraints than most forms of it. In effect mathematical imagination feels like it is convergent upon something that pre-exists the imagination. But that is just Platonism. Whether the target of convergence actually pre-exists in a separate realm or not, well, that's sort of just a way of talking, but as you say, we have to act like it is real, so why not just acknowledge it as real?
Sorry if I am clueless, I probably need to read your stuff in more detail.
I didn’t see Brouwer’s intuitionism mentioned in your previous post or the comments. Are you familiar with this notion of mentally constructed mathematical objects, and does your conceptual view differ significantly?
Re: platonic ontology vs. model theory, first-order logic is unable to capture “real” objects thanks to Gödel’s completeness and incompleteness theorems, but this might be an artifact of the formalization: the same isn’t necessarily true of second-order logic with the standard semantics, which has no completeness result.
In addition to Brouwer, I think you’d get something out of the talks and short expository papers of George Boolos and Edward Nelson.
Thanks for your comment. I had left intuitionism aside for simplicity, as most of my readers aren't familiar with it and also because it's trickier to categorize.
In a way, intuitionism *should* have been compatible with the conceptualist approach, and Brouwer had clearly been guided by a similar "intuition" of what was really at stake when he was doing math. I admire him and have a lot of sympathy for his stance.
Unfortunately, possibly because it came too early, intuitionism cristallized in a way that (in my view) completely misses the point:
- it failed to provide a clear account of what mathematical intuition is about, and never discussed how it is *formed* by the interaction with formal systems,
- instead, it instituted a dogma that equated intuitive math with constructive math and the rejection of the law of excluded middle.
While "intuitionist logic" is a valid and exciting domain of study, it is nothing but a formalist travesty of human intuition and its neural underpinnings.
Mathematicians and logicians have a hard time accepting the full conceptualist framework because it leads to refusing to formalize the connection between formal systems and intuition. Conceptualism regards semantics as a cognitive phenomenon that *cannot* be formalized—formalization is a tool for expressing and anchoring our intuition, not for modelling it. The "formalist travesty" through intuitionist logic of the intuitionist project might stem from this: for Brouwer and his school, accepting that intuition couldn't be formalized might have been a bridge too far.
PS: I didn't know about George Boolos and Edward Nelson, thanks for the reco.
I really appreciate the conceptualism view but on a philosophical level, I am somewhat uncertain whether it’s a solid counter-proposal to mathematical platonism, the metaphysical stance that mathematical objects exist independently of human cognition and culture. Essentially, I don’t see if this has anything to say about how these objects become part of human understanding or whether these are spiritual entities independent of physical reality. Is it possible Platonism has been widely misunderstood or that the real issue lies in its popular perception rather than the actual view embraced by self-declared mathematical Platonists?
To me, it seems like the Platonic view mainly stems from the need to distinguish the mathematical abstractions which are “probably-human-independent” universals from the “obviously-human-dependent” cognitive universals such as “beauty”, “worth” or “evil”; sustaining ancient spiritualistic traditions does not seem to me to be the primary motivation.
I understand that spiritualistic or esoteric worldviews can impede research and education and I advocate for neither. Still, I believe there might be room for Platonism because, in essence, it might be alluding to a subtle yet significant human need: to classify and distinguish between cognitive universals.
Writing as a psychoanalyst who uses dreaming as an ubiquitous biopsychosocial state of consciousness present through much of the animal kingdom, I think it would be helpful to demystify dreams and the dreamer without disregarding the many philosophical problems involved. It is not at all surprising, although it is amazing, that mathematicians have discovered or created ideational feats of imagination described in this article. Kekule, it is said discovered the circular structure of benzene in a dream of a circle-snake with its tail in its mouth. Freud’s magnum opus, some would say, is his Interpretation of Dreams. Each of the mathematicians cited lend Freudian support to the creativity of dreams and daydreams. However, psychoanalysis, despite its insights into the interaction of waking and sleeping mind, leaves many questions open to further investigation. It might be worthwhile considering one particular insight—that of two modes of human mentation—primary process, which is imagistic, symbolic, lightning fast in its motivation for relief and satisfaction, and secondary process, which is verbal, ordinarily logic-bound, prising delay over immediate gratification and presumably civilisation’s best friend in our march towards order, sanity, and predictability. Or as Oppenheimer said ‘Mathematics’ is “an immense enlargement of language, an ability to talk about things which in words would be simply inaccessible.”
I appreciate the emphasis on the creativity and imagination involved in doing maths, and the sense of almost divine "inspiration" striking in the process, but I think it's a mistake to completely deny platonism.
The way I see it, the mathematician is creatively exploring a space of possible patterns/structures. The possibility was always there, waiting to be realised, just as the physical possibility of building an aeroplane preceded the Wright brothers actually inventing one. But that platonic space of possibilities is explored via imagination and experimentation, not via special connection to a completely separate realm. The realness of the possibilities is why we can talk about mathematical discoveries, and why maths is more than just making stuff up. The fact that they are possibilities is why maths is a creative and imaginative exercise.
To be honest, I'm not sure I can see a way to make sense of the idea of "possibility" in general without introducing a degree of platonism.
I know that the nuance is tricky, but again Platonism is *methodologically* correct and I don't object to this version: you're entirely right to rely on it to "make sense" of abstract ideas.
The issue with *metaphysical* Platonism is that it obscures the active cognitive process that allows you to do that.
Would you say my account above obscures the cognitive process too? I can see how platonism often does (your diagram illustrates it really nicely), and I agree with the second diagram showing the role of imagination, but to me that act of imagination is precisely how we do interact with the space of possibilities.
What obscures the understanding of outsiders is the premise that mathematical objects "exist" but they can't see them. It makes them feel inadequate, and the situation is nonsensical.
Take complex numbers, for example. For me, they feel "real" and even "obvious": I "see" them, I "feel" them, it's like they've always been there.
But I know that I've worked at creating this stable mental image by actively engaging my imagination. So in a way I have "created" complex numbers in my mind. To someone who can't make sense of complex numbers, this bit of information is essential: there is a path from not seeing something to seeing it, and it's entirely normal to not see it at first, because it's not "before your eyes" in the same way an elephant is before your eyes.
I am slowly moving through philosophical ideas and thinkers but so slowly I can't say much about anything... With this caveat, I can't see why imagination can't be an access point to Platonic mathematical truths outside of us, where meaning and intelligibility are between us and the world. You're idea of 'projection of meaning' by us would seem to be open to a rather large critique of it's own. To Descartes and Galileo I would add Kant to people who have narrowed us into the current box we find ourselves in. It's not easy how self-generated projected meaning is meaningful unless it ties us to reality.
On metaphysical questions, it's virtually impossible to prove anything.
I can't prove that our imagination isn't a gateway to another world but, as would have said the great conceptualist William of Ockham, why would we make this assumption if it isn't needed?
Yes, hard to prove but not something easily avoided. More to think on for sure but it seems to me any account of anything will necessarily smuggle in some metaphysics. Especially if we're looking at the person, doing the maths.
It’s interesting the many religious systems of thought also teach that formalism is important and that the danger in departing from them is delusion. The practice of discernment is taken very seriously.
"There should be no secrets and no taboos regarding the underpinnings of rationality. The power of mathematics lies not in supernatural access to eternal truths, nor in unbridled imagination, but in the constant back-and-forth between an apparently sterile formalism and our continuing effort to make intuitive sense of it. " It is weird that you write this like a definitive statement. Is this not a metaphysical statement that can not be proved? So why write it like a stated truth instead of a claim?
(I am not debating whether or not this is the case, just how it is worded).
Every statement about the world contains unprovable metaphysical statements, if you decide to look for them.
Eg, what proves you that each elementary particle isn't remote-controlled by a deity who is deciding that it should stick to the laws of physics?
OK, this is an unfair straw man characterization of your valid objection, but still: my whole point is to challenge the millennial tradition of projecting Platonist metaphysics onto the practical activities that we call "rational thinking" and "mathematics".
Call me a materialist/physicalist if you want, and label all my statements within this if this makes you more comfortable.
But, from my perspective, what i'm simply doing is looking at language and semantics with maximally deflationary metaphysics.
No yeah I really liked your book. It was probably the most inspiring book I’ve read about math. And I do like your ideas about neuroplasticity and coupling it (is that the right word) to neuroscience. In particular I liked your focus on math as practice (comparing it to yoga at one point; funnily enough , as you know, one talks about ”yoga of grothendieck etc” in math at times). But I don’t think your position is incompatible with some kind of (broadly-defined) platonist position. Anyways I am not that interested in metaphysical discussions. But I do think (as far as I have thought about it) my own position is close to Joel David Hamkins (if that is his name) ”set-theoretic multiverse” or whatever he calls it. Anyways… =)
Sure. Just a correction, I don't think one perhaps says "yoga of grothendieck", but rather e.g. "yoga of motives".
Also, I actually do kind of like the analogy between mathematicians and shamans =). I saw some interview in which Barry Mazurs wife likened him to a shaman (if I recall corrrectly, see: https://www.youtube.com/watch?v=bEf26NFur3w&t=878s).
[But I am biased since I have had a long-lasting interest in psychedelics; and especially how one can learn about consciousness and its relation to the brain through their action (I like to think about them as particle-accelerators for the mind)]. =)
Edit: around timestamp 21:54 in the video in the link above.
Of course. My comment (in the book) about Grothendieck having invented his own yoga technique isn't a phrase that he ever used. It is my own metaphor, aimed at the general public, intentionally modelled on adjacent uses of the word that I knew many mathematicians would recognize:
- its fairly common association with motives
- the less well-known, broader and more casual use of the term in the early Grothendieck-Serre correspondence.
I do think the term does justice to the ethos of Grothendieck and Harvests and Sowings.
Language and human reasoning developed together, so language itself is a powerful tool for presenting ideas as if they came from human reason. Thus LLMs give us truthiness while being perfectly capable of lying.
Conversely, using nothing more than a rule for substitution, computers can leverage axioms into a powerful proof-checker of nearly any statement that can be expressed in a one-dimensional string of symbols. https://us.metamath.org/mpeuni/mmset.html
I share your belief in the universal distinction between signs and reality, but I can’t help feeling like mathematics operates on a different order.
New philosophical ideas often relitigate fundamental concepts like ‘being’, or ‘identity’, or ‘meaning’, shaking the whole symbolic tree and introducing new trade offs (‘our new explanation of X is problematic for our existing understanding of Y’).
Mathematics (and I am no mathematician!) just doesn’t seem to such an accounting problem:
1. The basic operators of formal systems remain largely unchanged;
2. Formal systems seem to compete with each other far less over what those operators and terms mean, in a given context;
3. Through maths, we can articulate predictions about real-world behaviour we haven’t yet observed but turn out to be correct, which lends maths this sense of ‘being out there’
Another way of saying it, I suppose, is that mathematics seems to be converging on a 1:1 description of reality in ways that other symbolic systems aren’t: new proofs tell us new things without compromising the old.
There is indeed a specificity of mathematics: it is based on using consistent formal systems as a way to ground truth. This is what protects it from the "accounting problem", and this is what creates the cumulative effect: symbolic systems are stable.
Mathematics is based on renegotiating the deal between signs and meaning: in normal human language, there's a shared experience of perceived "meaning" that is then captured in words, then transcribed in signs; when doing math, you force yourself to recalibrate your perception of "meaning" based on what signs actually tell you. This is different cognitive pathway, which explains why it's so hard to teach.
"Imagination shouldn’t be a dirty word." It isn't, who said it was? But you insist that if imagination is involved, then it's not directly accessing "objective" reality, and that to think otherwise is some weird mystical bent. I disagree. You use your imagination every time you conceive of other human beings as genuinely conscious entities. Just because you have to imagine it doesn't mean it isn't objectively real.
"Of course, no-one has a clue as to how mathematicians’ brains can magically “access” the ethereal world of ideal entities." Yes we do. We use our imagination.
I'm not denying that there is something that deserves to be called "reality".
I'm challenging, for example, the notion that the "sets" supposedly described by ZFC theory are "objectively real".
It's quite interesting, by the way, to notice that ZFC theory doesn't contain the word "set" except on the title page. And, if you stick to the formalist "objectivity", the symbol ∀ doesn't even mean "for all"—it's just another meaningless symbol in the theory.
How do you go from there to your idea of a "set"? How could this idea be "objective" in any way, given that ZFC isn't even complete?
Through this example, I'm trying to illustrate what is for me the most compelling point: the "truth" value of a statement (ie, the central notion in mathematics) cannot be more objective than the semantics of this statement; but semantics itself is a cognitive phenomenon. While syntax is objective, semantics CANNOT be objective (unless it is objectified by model theory as another syntactic expression, but this isn't the true semantics that Platonists care about.)
Has a philosophy, major and lifelong reader, I’m shocked that I’ve never thought much about the philosophy of Mathematics. You made it not only interesting but essential.
Wonderful. You might be interested in this https://omniorthogonal.blogspot.com/2013/05/the-opposite-of-mathematics.html which also talks about the Bob Thomason / ghost of Tom Trobaugh collaboration.
I'm a little confused by what the philosophical stakes are here. I'm perfectly happy to acknowledge the role of imagination in mathematics, but I don't quite see how that refutes or contradicts Platonism.
Imagination is not, in general, producing just any old thing. It has constraints, even dreams and fictions have constraints. Mathematical imagination has sharper constraints than most forms of it. In effect mathematical imagination feels like it is convergent upon something that pre-exists the imagination. But that is just Platonism. Whether the target of convergence actually pre-exists in a separate realm or not, well, that's sort of just a way of talking, but as you say, we have to act like it is real, so why not just acknowledge it as real?
Sorry if I am clueless, I probably need to read your stuff in more detail.
I didn’t see Brouwer’s intuitionism mentioned in your previous post or the comments. Are you familiar with this notion of mentally constructed mathematical objects, and does your conceptual view differ significantly?
Re: platonic ontology vs. model theory, first-order logic is unable to capture “real” objects thanks to Gödel’s completeness and incompleteness theorems, but this might be an artifact of the formalization: the same isn’t necessarily true of second-order logic with the standard semantics, which has no completeness result.
In addition to Brouwer, I think you’d get something out of the talks and short expository papers of George Boolos and Edward Nelson.
Thanks for your comment. I had left intuitionism aside for simplicity, as most of my readers aren't familiar with it and also because it's trickier to categorize.
In a way, intuitionism *should* have been compatible with the conceptualist approach, and Brouwer had clearly been guided by a similar "intuition" of what was really at stake when he was doing math. I admire him and have a lot of sympathy for his stance.
Unfortunately, possibly because it came too early, intuitionism cristallized in a way that (in my view) completely misses the point:
- it failed to provide a clear account of what mathematical intuition is about, and never discussed how it is *formed* by the interaction with formal systems,
- instead, it instituted a dogma that equated intuitive math with constructive math and the rejection of the law of excluded middle.
While "intuitionist logic" is a valid and exciting domain of study, it is nothing but a formalist travesty of human intuition and its neural underpinnings.
Mathematicians and logicians have a hard time accepting the full conceptualist framework because it leads to refusing to formalize the connection between formal systems and intuition. Conceptualism regards semantics as a cognitive phenomenon that *cannot* be formalized—formalization is a tool for expressing and anchoring our intuition, not for modelling it. The "formalist travesty" through intuitionist logic of the intuitionist project might stem from this: for Brouwer and his school, accepting that intuition couldn't be formalized might have been a bridge too far.
PS: I didn't know about George Boolos and Edward Nelson, thanks for the reco.
I really appreciate the conceptualism view but on a philosophical level, I am somewhat uncertain whether it’s a solid counter-proposal to mathematical platonism, the metaphysical stance that mathematical objects exist independently of human cognition and culture. Essentially, I don’t see if this has anything to say about how these objects become part of human understanding or whether these are spiritual entities independent of physical reality. Is it possible Platonism has been widely misunderstood or that the real issue lies in its popular perception rather than the actual view embraced by self-declared mathematical Platonists?
To me, it seems like the Platonic view mainly stems from the need to distinguish the mathematical abstractions which are “probably-human-independent” universals from the “obviously-human-dependent” cognitive universals such as “beauty”, “worth” or “evil”; sustaining ancient spiritualistic traditions does not seem to me to be the primary motivation.
I understand that spiritualistic or esoteric worldviews can impede research and education and I advocate for neither. Still, I believe there might be room for Platonism because, in essence, it might be alluding to a subtle yet significant human need: to classify and distinguish between cognitive universals.
Edit: Comment was too long, shortened.
Writing as a psychoanalyst who uses dreaming as an ubiquitous biopsychosocial state of consciousness present through much of the animal kingdom, I think it would be helpful to demystify dreams and the dreamer without disregarding the many philosophical problems involved. It is not at all surprising, although it is amazing, that mathematicians have discovered or created ideational feats of imagination described in this article. Kekule, it is said discovered the circular structure of benzene in a dream of a circle-snake with its tail in its mouth. Freud’s magnum opus, some would say, is his Interpretation of Dreams. Each of the mathematicians cited lend Freudian support to the creativity of dreams and daydreams. However, psychoanalysis, despite its insights into the interaction of waking and sleeping mind, leaves many questions open to further investigation. It might be worthwhile considering one particular insight—that of two modes of human mentation—primary process, which is imagistic, symbolic, lightning fast in its motivation for relief and satisfaction, and secondary process, which is verbal, ordinarily logic-bound, prising delay over immediate gratification and presumably civilisation’s best friend in our march towards order, sanity, and predictability. Or as Oppenheimer said ‘Mathematics’ is “an immense enlargement of language, an ability to talk about things which in words would be simply inaccessible.”
I appreciate the emphasis on the creativity and imagination involved in doing maths, and the sense of almost divine "inspiration" striking in the process, but I think it's a mistake to completely deny platonism.
The way I see it, the mathematician is creatively exploring a space of possible patterns/structures. The possibility was always there, waiting to be realised, just as the physical possibility of building an aeroplane preceded the Wright brothers actually inventing one. But that platonic space of possibilities is explored via imagination and experimentation, not via special connection to a completely separate realm. The realness of the possibilities is why we can talk about mathematical discoveries, and why maths is more than just making stuff up. The fact that they are possibilities is why maths is a creative and imaginative exercise.
To be honest, I'm not sure I can see a way to make sense of the idea of "possibility" in general without introducing a degree of platonism.
Thanks for your comment.
I know that the nuance is tricky, but again Platonism is *methodologically* correct and I don't object to this version: you're entirely right to rely on it to "make sense" of abstract ideas.
The issue with *metaphysical* Platonism is that it obscures the active cognitive process that allows you to do that.
Thanks for your reply :)
Would you say my account above obscures the cognitive process too? I can see how platonism often does (your diagram illustrates it really nicely), and I agree with the second diagram showing the role of imagination, but to me that act of imagination is precisely how we do interact with the space of possibilities.
Not your account itself.
What obscures the understanding of outsiders is the premise that mathematical objects "exist" but they can't see them. It makes them feel inadequate, and the situation is nonsensical.
Take complex numbers, for example. For me, they feel "real" and even "obvious": I "see" them, I "feel" them, it's like they've always been there.
But I know that I've worked at creating this stable mental image by actively engaging my imagination. So in a way I have "created" complex numbers in my mind. To someone who can't make sense of complex numbers, this bit of information is essential: there is a path from not seeing something to seeing it, and it's entirely normal to not see it at first, because it's not "before your eyes" in the same way an elephant is before your eyes.
I am slowly moving through philosophical ideas and thinkers but so slowly I can't say much about anything... With this caveat, I can't see why imagination can't be an access point to Platonic mathematical truths outside of us, where meaning and intelligibility are between us and the world. You're idea of 'projection of meaning' by us would seem to be open to a rather large critique of it's own. To Descartes and Galileo I would add Kant to people who have narrowed us into the current box we find ourselves in. It's not easy how self-generated projected meaning is meaningful unless it ties us to reality.
On metaphysical questions, it's virtually impossible to prove anything.
I can't prove that our imagination isn't a gateway to another world but, as would have said the great conceptualist William of Ockham, why would we make this assumption if it isn't needed?
Yes, hard to prove but not something easily avoided. More to think on for sure but it seems to me any account of anything will necessarily smuggle in some metaphysics. Especially if we're looking at the person, doing the maths.
It’s interesting the many religious systems of thought also teach that formalism is important and that the danger in departing from them is delusion. The practice of discernment is taken very seriously.
Forgot to say great thought-provoking post with fascinating historical accounts!
"There should be no secrets and no taboos regarding the underpinnings of rationality. The power of mathematics lies not in supernatural access to eternal truths, nor in unbridled imagination, but in the constant back-and-forth between an apparently sterile formalism and our continuing effort to make intuitive sense of it. " It is weird that you write this like a definitive statement. Is this not a metaphysical statement that can not be proved? So why write it like a stated truth instead of a claim?
(I am not debating whether or not this is the case, just how it is worded).
Every statement about the world contains unprovable metaphysical statements, if you decide to look for them.
Eg, what proves you that each elementary particle isn't remote-controlled by a deity who is deciding that it should stick to the laws of physics?
OK, this is an unfair straw man characterization of your valid objection, but still: my whole point is to challenge the millennial tradition of projecting Platonist metaphysics onto the practical activities that we call "rational thinking" and "mathematics".
Call me a materialist/physicalist if you want, and label all my statements within this if this makes you more comfortable.
But, from my perspective, what i'm simply doing is looking at language and semantics with maximally deflationary metaphysics.
No yeah I really liked your book. It was probably the most inspiring book I’ve read about math. And I do like your ideas about neuroplasticity and coupling it (is that the right word) to neuroscience. In particular I liked your focus on math as practice (comparing it to yoga at one point; funnily enough , as you know, one talks about ”yoga of grothendieck etc” in math at times). But I don’t think your position is incompatible with some kind of (broadly-defined) platonist position. Anyways I am not that interested in metaphysical discussions. But I do think (as far as I have thought about it) my own position is close to Joel David Hamkins (if that is his name) ”set-theoretic multiverse” or whatever he calls it. Anyways… =)
Best,
Ben
Great :) !
And, yes, my position isn't incompatible with some kind of (broadly-defined) platonist position... Broadly-defined platonism is unfalsifiable.
Sure. Just a correction, I don't think one perhaps says "yoga of grothendieck", but rather e.g. "yoga of motives".
Also, I actually do kind of like the analogy between mathematicians and shamans =). I saw some interview in which Barry Mazurs wife likened him to a shaman (if I recall corrrectly, see: https://www.youtube.com/watch?v=bEf26NFur3w&t=878s).
[But I am biased since I have had a long-lasting interest in psychedelics; and especially how one can learn about consciousness and its relation to the brain through their action (I like to think about them as particle-accelerators for the mind)]. =)
Edit: around timestamp 21:54 in the video in the link above.
Of course. My comment (in the book) about Grothendieck having invented his own yoga technique isn't a phrase that he ever used. It is my own metaphor, aimed at the general public, intentionally modelled on adjacent uses of the word that I knew many mathematicians would recognize:
- its fairly common association with motives
- the less well-known, broader and more casual use of the term in the early Grothendieck-Serre correspondence.
I do think the term does justice to the ethos of Grothendieck and Harvests and Sowings.
Language and human reasoning developed together, so language itself is a powerful tool for presenting ideas as if they came from human reason. Thus LLMs give us truthiness while being perfectly capable of lying.
Conversely, using nothing more than a rule for substitution, computers can leverage axioms into a powerful proof-checker of nearly any statement that can be expressed in a one-dimensional string of symbols. https://us.metamath.org/mpeuni/mmset.html
I share your belief in the universal distinction between signs and reality, but I can’t help feeling like mathematics operates on a different order.
New philosophical ideas often relitigate fundamental concepts like ‘being’, or ‘identity’, or ‘meaning’, shaking the whole symbolic tree and introducing new trade offs (‘our new explanation of X is problematic for our existing understanding of Y’).
Mathematics (and I am no mathematician!) just doesn’t seem to such an accounting problem:
1. The basic operators of formal systems remain largely unchanged;
2. Formal systems seem to compete with each other far less over what those operators and terms mean, in a given context;
3. Through maths, we can articulate predictions about real-world behaviour we haven’t yet observed but turn out to be correct, which lends maths this sense of ‘being out there’
Another way of saying it, I suppose, is that mathematics seems to be converging on a 1:1 description of reality in ways that other symbolic systems aren’t: new proofs tell us new things without compromising the old.
There is indeed a specificity of mathematics: it is based on using consistent formal systems as a way to ground truth. This is what protects it from the "accounting problem", and this is what creates the cumulative effect: symbolic systems are stable.
Mathematics is based on renegotiating the deal between signs and meaning: in normal human language, there's a shared experience of perceived "meaning" that is then captured in words, then transcribed in signs; when doing math, you force yourself to recalibrate your perception of "meaning" based on what signs actually tell you. This is different cognitive pathway, which explains why it's so hard to teach.
thank you for continuing to publish your thoughts and create these discussions.
"Imagination shouldn’t be a dirty word." It isn't, who said it was? But you insist that if imagination is involved, then it's not directly accessing "objective" reality, and that to think otherwise is some weird mystical bent. I disagree. You use your imagination every time you conceive of other human beings as genuinely conscious entities. Just because you have to imagine it doesn't mean it isn't objectively real.
"Of course, no-one has a clue as to how mathematicians’ brains can magically “access” the ethereal world of ideal entities." Yes we do. We use our imagination.
I'm not denying that there is something that deserves to be called "reality".
I'm challenging, for example, the notion that the "sets" supposedly described by ZFC theory are "objectively real".
It's quite interesting, by the way, to notice that ZFC theory doesn't contain the word "set" except on the title page. And, if you stick to the formalist "objectivity", the symbol ∀ doesn't even mean "for all"—it's just another meaningless symbol in the theory.
How do you go from there to your idea of a "set"? How could this idea be "objective" in any way, given that ZFC isn't even complete?
Through this example, I'm trying to illustrate what is for me the most compelling point: the "truth" value of a statement (ie, the central notion in mathematics) cannot be more objective than the semantics of this statement; but semantics itself is a cognitive phenomenon. While syntax is objective, semantics CANNOT be objective (unless it is objectified by model theory as another syntactic expression, but this isn't the true semantics that Platonists care about.)
Wow I loved reading this. Thanks!
Has a philosophy, major and lifelong reader, I’m shocked that I’ve never thought much about the philosophy of Mathematics. You made it not only interesting but essential.
Thanks!