by CJ Uberroth

As a high school math teacher, I have had my fair share of interesting conversations, ranging from my students’ favorite music to my addiction to American Spirit cigarettes. Sure, some may say that I shouldn’t speak with students about things like alcohol and nicotine but I can’t imagine a stupider objection. How else are they supposed to learn the real reasons one shouldn’t smoke? I can promise that you that the only conversations about “substances” that a teenager will listen to begin with the following sentence: “Well *of course* I smoked pot, but listen for a second and I’ll explain that the only reason I’m telling you not to is because you’re doing it stupidly.” But, I digress. The point simply is that I have interesting conversations with my teenage students. And the most interesting of all time may have occurred this past week. While I was writing the work for a related-rates problem on the board of my calculus class, I heard the most amazing word: “Why?”

This wasn’t some wasted breath. Nor was it some stray remark from Klark, one of my (brilliant) asshole seventeen year-olds, thinking he’d be funny by disrupting my class. This question came from one of the more thoughtful, bright seniors and was quickly followed up with a somewhat intense clarification:

“I mean, who came up with this? Why did they do it? Why was someone spending their time doing this?”

What a question! I felt like an only child walking into his living-room on Christmas morning. “Are all of these for me?” I was actually a bit stunned, but fearing that she might lose interest or Klark might strike without warning, I jumped into the sort of discussion I’ve only been able to have with a glass of Scotch and a cigarette and with people who already knew every idea I could possibly throw at them.

“Well,” I began, “it depends how far you want to consider this. As we all know – or at least should know – most theories in calculus were formed by Newton and Leibniz. The reasoning behind those theories will be tough to nail down. Each of them probably…”

And then she cut me off. Not because of boredom or regret for asking the question, but for even *further* clarification.

“I guess what I really meant was, how? How did this all come to be?”

At this point her friends were hanging on the conversation as well. Amazing. Here I had five or six high school seniors, in the middle of small-town Texas, asking about the philosophical theories behind mathematics. I figured I wouldn’t bother telling my colleagues, as I knew they wouldn’t believe it.

Then Plato hit me. For those of you who are unaware, philosophy of mathematics has arguably three branches of thought. There are the science-minded people who simply view mathematics as empirical and nothing more. This camp includes people from all flocks including empiricists and various other schools. Boring! I mean, sure, this is arguably the easiest concept and the most practical but it’s, well, boring. (Not to mention, probably wrong.)

Platonism describes those (including a younger version of myself) who believe that mathematics has its own independent existence and is waiting to be discovered and understood. What a wonderful idea! Imagine, the Pythagorean Theorem actually is an ancient Grecian monster, akin to Theseus’ minotaur, trained by Pythagoras in 500 BC to be available to all humankind for eternity. A tale worth telling, that! Metaphors aside, the idea is powerful, suggesting that mathematics exists as a thing of its own, with its own distinctive nature.

So, I jumped in head first, explaining that what seems like an absurd encyclopedia of random rules in calculus is really all tied together. Without the Intermediate Value Theorem, you have no Mean Value Theorem. Without the Fundamental Theorem of Calculus, you have no derivative. I was relentless. I even began to start showing the ties from integrals all the way down to Euclid’s Elements. Showing that everything was intertwined in these subtle ways really had their attention. But, it wasn’t meant to be.

Just as I was explaining the importance and acceptance of Euclid’s Elements, the fundamental axioms on which all of our basic mathematics are based, Klark struck. Who else but the smartest student in the class will point out the antithesis of this theory? If you’ve been following closely, you’ll notice that I left off the third of the three branches of mathematical philosophy. This was not by accident. I didn’t want to mention the final idea. But sure enough, Klark had thought of it.

Fictionalism, as the name suggests, is the view that mathematics is nothing more than a human invention, reflecting the human desire to find order in chaos. I hate it. I mean, why would someone strip away the beautiful idea that statements like “if the sum of two squares is equal to a third square, then the angle opposite the largest value in a triangle constructed from these values will be 90 degrees” describes a remarkable, delightful independent reality? But sure enough, Klark did it.

“That doesn’t make sense,” he sniveled. “How can mathematics ‘exist’ (he actually used air-quotes), if it all boils down to some set of rules that Euclid invented that have to be assumed to be true?”

I wanted to cry or scream or maybe smack him upside the head. I really can’t remember the exact feeling. But, I had to settle for a quiet nod of acknowledgment. He was right. I knew it, my students knew it, and, worst of all, *he* knew it. So after my twenty minutes of blowing my students’ minds with connected theories, sublime equalities, and incredible identities I had to come clean.

“Yes, Klark, I left one theory off my list.” I have long realized (grudgingly) that Fictionalism is the best philosophical account of mathematics. I’ve even developed a pseudo-proof with an old friend of mine. If I asked you what came after 10 in the sequence 4, 7, 10, … you would say “13.” But what if I told you it was 37? How would you argue with me? I never told you that the sequence was the progression of adding three each time. We could be following any rule you could imagine. Hell, I never even told you that the sequence had to continue, which means the answer could have been “nothing.” But we want it to be 13 don’t we? To me, that is enough to prove Fictionalism drips with the truth (along with the blood of my Platonic youth).

I couldn’t believe it. Thwarted by the student who sings “All Star” by Smash Mouth in the back of my class just to piss me off. Yes, I know Platonism is false, but hear me out. I want to help educate a group of people that will go out into the world and become scientists and mathematicians, whether by trade or by hobby. Imagine Dan Kaufman and I are standing in front of you. Beside each of us is a boulder that we ask you to roll up a hill. Dan offers you ten days and nights at his private resort on the Miami Beach, with all expenses paid, and I tell you that I’m gonna’ kick it back down the hill the minute you’re done. Who will you do it for? How devastating to a young learner is Fictionalism? Realizing that even if you master everything that mathematics has to offer, it is ultimately an exercise in futility, as it all could have been entirely different; or could be rewritten next week; that none of it describes anything that’s *really there*.

Having explained all three theories to my students, now, and feeling quite deflated, I wrapped up with a snarky quip at Klark. It was over. My students would continue in mathematics just until it was done paying off for their future careers in medicine or engineering or whatever. Ugh, it hurt, that feeling of failure. To think I was so close to convincing a group of young people that mathematics was beautiful and profound. Something more than a mere tool. The bell rang to signal us to class shortly after that. As everyone packed up, the student who had asked the question that started this whole thing stopped at my desk.

“You know, Uberroth,” she said, “that, Platonism thing…?”

“Yeah?” I replied.

“It’s *way* more of a reason to do math than what Klark said. Thanks!” And then she left.

I leaned back in my chair, with a satisfied sigh. I had made a believer out of at least one student that day. Then, I contemplated what practical joke I should play on Klark during the next day’s class. I swore to myself that it would be brutal.

I don’t know about that. I think I was a fictionalist from the get-go. Platonism never made sense to me. But mathematics still seemed a thing of beauty. No devastation here.

And there, I completely disagree. I never saw mathematics as about what is. Rather, I always saw it as methodological. For the Pythagorean theorem, it was the proof, with its clever constructions, that seemed important. The actual conclusion about sums of squares seemed more mundane.

That the mathematical entities are merely useful fictions, does not invalidate the effectiveness of the methodology that is used.

Great post- thank you. I hope you found a really excellent prank to play on your brilliant student.

I’m interested in the conversation that I’m sure will ensue because of this notion that education strips away mystery and intrigue. As a musician, I’ve often (quietly, privately) lamented that I don’t really enjoy many concerts- I understand how the music is made, and because of my many years of training, I often hear every mistake. Even if there are no mistakes, I’m listening for them, which also ruins the experience. I hear when flute players are using “cheat” fingerings, when they crack notes, and when their technique is generally not up to par. Hearing an orchestra tune up, which used to give me chills (how ancient and mysterious that sound was!) now often makes me roll my eyes when I hear a brass player can’t quite find the A.

But mystery is a mobilizer- the ancient, the esoteric, the connections are all reasons to pursue something, whether it’s music or math. Many times I feel like I have chosen to reject the theory that seems correct in favor of the one that I love. And why not? Who says that we are not allowed to create meaning where we can? There must be some third category other than “likely not” and “likely”: maybe it’s “likely not but I choose to believe it anyway.”

Welcome Craig (“CJ”),

I enjoyed the essay, but the premise doesn’t resonate for me. The idea that mathematics is a step by step ( open ended ) product of collective human creativity is much more inspiring to me than the idea it might be somehow already sitting out there in some finalized form waiting to be discovered.

Why should we assume invention – especially invention that opens up new ways of seeing and interacting with world – double especially invention that depends on engagement with prior creative human inventiveness – as less than inspiring than bumping into something pre-defined for us? Am I really an outlier to think this way?

CJ,

“

To me, that is enough to prove Fictionalism drips with the truth (along with the blood of my Platonic youth).”But what if Platonism really was true? Mathematics would still look the same. You would still reach this conclusion but you would be wrong. So what have you shown, other than you want to believe in Fictionalism?

But wait, I am being rude. This is no way to greet a new guest. Welcome and congratulations on a lovely essay. I hope we see more of Klark and your unnamed student, who deserves a leading role. Perhaps her reaction is an indicator of the truth. Perhaps our intuitive reaction to the beauty of mathematics and its remarkable fit to reality points to a Platonic truth.

Welcome CJ.

First of all, this was a joy to read. Hilarious and heartfelt at the same time.

Second, I agree with you and disagree with the critics. I too was a Platonist in my earlier years, precisely for the reason that I found magical the idea that there was an entire universe beyond the realm of perception; a strange, almost impossible to imagine place in which there were things that weren’t things in the way one normally thinks of things. Unbeknownst to me, this represented the remains of my youthful imagination, which, now that I was more intellectually sophisticated, was manifesting itself via Platonism, rather than through overt science fiction fantasies.

The process of coming to no longer think this was part and parcel of my coming to sit more comfortably with an adult frame of mind, but as a result, it also was part and parcel of the demystification of the world which, while largely good, is still, in part, something to lament.

Seth,

“

Why should we assume invention …as less than inspiring than bumping into something pre-defined for us?”We are explorers that thrill at discovery, we are creators that delight in making the new and we are performers excited by the roles we play.

Not all of us can excel as creators or performers but all of us can experience the delight of discovery, even if it is only through enjoying the discoveries of others, because that then becomes our own discovery.

Labnut,

I agree with your comment. I was speaking of ‘invention’ in the sense of collective human activity, and I think all inventions result from integrating discoveries of what others have done. That is what I find inspiring with what it implies for human potential. I may not have the talent to invent something genuinely novel and influential myself, but each time I discover something novel to my understanding, and that discovery allows me to see the world slightly differently I find it very inspiring.

Margret’s comment is giving me some pause.

I think when we gain a certain virtuosity or fluency in some medium it becomes tedious to engage with that medium at lower levels of fluency. I know when I learned how to juggle 5 balls and became proficient at that level it became tedious. Then I adapted by learning to alter the shape of the pattern while listening to music and it became fresh an engaging again. Then I eventually learned six balls. This may relate to the idea of working and engaging at skill level appropriate for our ability or fluency as suggested by flow researcher Mihály Csíkszentmihályi.

I find it interesting that what enchants us can vary so much from one person to the next. I’m certainly not arguing that my view of enchantment is the correct one.

Mirowley,

“But mystery is a mobilizer”

That’s one of the fascinating things in mathematics: the mystery never disappears if you’re sensitive to it.

CJ,

beautiful piece. My partner is a mathematician and a teacher with 30 yrs experience. One of the greatest joys in her life is having students who are sensitive to the aesthetics and the mystery of mathematics.

And mathematics is still mysterious to her too. She knows the stuff she teaches from A to Z, but it still happens that she comes home and says: today something I don’t understand happened. I gave my students an exercise that involves a lengthy calculation, and there was this student who had this crazy idea and he (or she) solved it in three lines. What’s going on?

(She never hesitates to admit to her students that she doesn’t know – in fact, the greatest joy in her life probably is having a student who is better at mathematics than she is).

We both have a very solid background in mathematics (I studies physics) but often it takes us one or two hours to analyse this crazy idea, to find out what’s going on, why it worked – or why it was a lucky shot.

After 30 yrs of teaching! Marvellous.

On the other hand, I must admit that we are both completely insensitive to questions about Platonism, Fictionalism etc. We both wonder why somebody could be interested in these questions while you could spend your time doing something infinitely more interesting – namely mathematics.

CJ, EJ says welcome, and thanks for bringing a light touch to a subject which, in other venues, I have seen worked into the grave to nobody’s benefit.

labnut,

“But what if Platonism really was true? Mathematics would still look the same.” I have a sense that this is the problem, because I suspect that mathematics would not look the same. For instance, would we have made the transition to non-euclidean geometry if we had a Platonistic commitment to Euclid (as I think some actually did at the time…). Also, there’s an awful lot of calculating done in physics and in computer programing that is generative rather than ‘discovery’-based. But I admit I have not the expertise in mathematics to argue this point; my own acceptance of ‘Fictionalism’ is derived from Pragmatist commitments. Mathematics can be a discovery – of the human capacity for invention. Platonism seems grounded in the assumption that a mathematical truth, since purely deductively arrived at, is atemporally true – it therefore must always have been true, even before no one thought of it. This presumes a Realist notion of truth that I am uncomfortable with. Western harmonics are built on what is a mathematical structure; but not every music partakes of the same structure. Rather, scales seem to have been developed to meet the needs of the cultures in which they appear. It seems to me to ask too much to ask that all these differing structures were always ‘true’ prior to their usefulness, since their very reason for existence is exactly their usefulness in composing music appealing to different audiences in the differing cultures.

Hi CJ. Nice piece, and I hope for more on this general topic.

I’m not sure whether my views on the philosophy of math are settled or not. I think many of the problems and differences you encounter are terminological. Clearly, there are substantively different views on offer, but it’s always a shame if people with compatible views perceive themselves (because of the way their respective views are labelled) as being in different – and opposing – camps.

Margaret

I agree that “mystery is a mobilizer”; but this bit I want to respond to:

“Many times I feel like I have chosen to reject the theory that seems correct in favor of the one that I love. And why not? Who says that we are not allowed to create meaning where we can? There must be some third category other than “likely not” and “likely”: maybe it’s “likely not but I choose to believe it anyway.” ”

Sure, we create meaning where we can, but this freedom is not absolute and depends, I would say (and I’m sure you would too) on the sort of belief we are talking about.

Also – forgive me if this sounds unimaginative – but how can you believe something which you also believe is less likely than an alternative view to be the case? (Serious question. I am not trying to be tricksy.)

Margaret: “

education strips away mystery and intrigue … But mystery is a mobilizer”Dan-K: “

part and parcel of the demystification of the world which, while largely good, is still, in part, something to lament.”Seth: “

each time I discover something novel to my understanding, and that discovery allows me to see the world slightly differently I find it very inspiring … I find it interesting that what enchants us can vary so much from one person to the next”There is a common thread here, that of seeing enchantment in the world and of the later loss of the feeling of enchantment. For Margaret this was the result of a deep understanding of techniques in musical performances. For Dan-K this was part and parcel of an adult frame of mind that demystifies the world. For Seth demystification came with proficiency. For Platonists there is enchantment in mathematics while Fictionalists deny this.

Is our capacity for feeling enchantment a reflection of some property of the world? Or is it just a psychological condition that is lost with maturity, growing knowledge of the details or methodology, or greater proficiency?

When we fall in love we become enchanted and we will remember this as our deepest, most meaningful experience ever. But then familiarity and intimate knowledge, for many, breaks the spell of enchantment, or the other person departs, breaking the enchantment. And then we discover how painful the loss of enchantment can be. In our loves and in our lives, we seek, and indeed we need enchantment. A great artist is one with a special faculty for perceiving enchantment in the world and then for breathing the spell of enchantment over us, so that we too may feel the enchantment. We attend to art because in art we re-discover enchantment.

But is this enchantment real? For me it is. Tonight I will go to Mass. As I kneel there, pray and take part in the sacred liturgies I will feel the deep enchantment that permeates and sustains the world. As I look around me at my fellow congregants I will see in the total immersion of their devotion that they are also under the spell of enchantment. I will come away renewed, invigorated and ennobled.

As I pray the

Daily Examen, I review my day to reflect on the enchantment I saw in the world during the day.As a photographer I try to capture moments of enchantment that I see in the world around me, so that I may remember it or show it to others, so that they can enjoy that moment of wonder.

Most of you will deny the reality of enchantment, maintaining it is only a psychological condition, but that does not matter. What matters is that we need enchantment in our lives, whatever its source. We are enriched when we find it and we are impoverished when we lose it.

One of the most beautiful descriptions of enchantment can be found in chapter 7 of Kenneth Grahame’s

The Wind in the Willows.http://www.cleavebooks.co.uk/grol/grahame/wind07.htm

To fully grasp this one should read the entire chapter. Below are excerpts:-

”

Their old haunts greeted them again in other raiment, as if they had slipped away and put on this pure new apparel and come quietly back, smiling as they shyly waited to see if they would be recognised again under it.

…

‘It’s gone!’ sighed the Rat, sinking back in his seat again. ‘So beautiful and strange and new. Since it was to end so soon, I almost wish I had never heard it. For it has roused a longing in me that is pain, and nothing seems worth while but just to hear that sound once more and go on listening to it for ever. No! There it is again!’ he cried, alert once more. Entranced, he was silent for a long space, spellbound.

‘Now it passes on and I begin to lose it,’ he said presently. ‘O Mole! the beauty of it! The merry bubble and joy, the thin, clear, happy call of the distant piping! Such music I never dreamed of, and the call in it is stronger even than the music is sweet! Row on, Mole, row! For the music and the call must be for us.’

The Mole, greatly wondering, obeyed. ‘I hear nothing myself,’ he said, ‘but the wind playing in the reeds and rushes and osiers.’

[Typical of the Reductionists, Scientismists and Fictionalists]

The Rat never answered, if indeed he heard. Rapt, transported, trembling, he was possessed in all his senses by this new divine thing that caught up his helpless soul and swung and dandled it, a powerless but happy infant in a strong sustaining grasp.

…

Breathless and transfixed the Mole stopped rowing as the liquid run of that glad piping broke on him like a wave, caught him up, and possessed him utterly. He saw the tears on his comrade’s cheeks, and bowed his head and understood. For a space they hung there, brushed by the purple loose-strife that fringed the bank; then the clear imperious summons that marched hand-in-hand with the intoxicating melody imposed its will on Mole, and mechanically he bent to his oars again. And the light grew steadily stronger, but no birds sang as they were wont to do at the approach of dawn; and but for the heavenly music all was marvellously still.

…

Never had they noticed the roses so vivid, the willow-herb so riotous, the meadow-sweet so odorous and pervading.

…

‘This is the place of my song-dream, the place the music played to me,’ whispered the Rat, as if in a trance. ‘Here, in this holy place, here if anywhere, surely we shall find Him!’

Then suddenly the Mole felt a great Awe fall upon him, an awe that turned his muscles to water, bowed his head, and rooted his feet to the ground. It was no panic terror— indeed he felt wonderfully at peace and happy— but it was an awe that smote and held him and, without seeing, he knew it could only mean that some august Presence was very, very near. With difficulty he turned to look for his friend. and saw him at his side cowed, stricken, and trembling violently. And still there was utter silence in the populous bird-haunted branches around them; and still the light grew and grew.

Perhaps he would never have dared to raise his eyes, but that, though the piping was now hushed, the call and the summons seemed still dominant and imperious. He might not refuse, were Death himself waiting to strike him instantly, once he had looked with mortal eye on things rightly kept hidden. Trembling he obeyed, and raised his humble head; and then, in that utter clearness of the imminent dawn, while Nature, flushed with fulness of incredible colour, seemed to hold her breath

…

All this he saw, for one moment breathless and intense, vivid on the morning sky; and still, as he looked, he lived; and still, as he lived, he wondered.

“

EJ,

“

I suspect that mathematics would not look the same.”You may suspect it, but we need something more substantive than that.

“

would we have made the transition to non-euclidean geometry if we had a Platonistic commitment to Euclid”Platonism is not a commitment to Euclidean geometry. It is a commitment to a truth that needs to be discovered, whatever the form it takes.

“

Also, there’s an awful lot of calculating done in physics and in computer programing that is generative rather than ‘discovery’-based”Platonism does not exclude generative computing. Generative computing can be a means of building on Platonism.

“

Mathematics can be a discovery – of the human capacity for invention”Yes, the human mind is endlessly creative. To use an example there are endless ways of expressing creativity through landscape paintings. No two landscapes need be the same and they are dependent on the person’s mind.

But this is completely unlike mathematics where, in a given situation, we find truth that seems to be independent of the mathematician’s mind.

“

Platonism seems grounded in the assumption that a mathematical truth, since purely deductively arrived at, is atemporally true”It is more than atemporarily true, it is mind independent.

“

my own acceptance of ‘Fictionalism’ is derived from Pragmatist commitments.”Yes, metaphysical a prioris are what really lies at the heart of opposition to Platonism.

“

This presumes a Realist notion of truth that I am uncomfortable with”Discomfort should not result in rejection but rather in exploration. Rejection needs far stronger grounds. A clue to the discomfort can be found in the provocative title of Mario Livio’s book – ‘

Is God a Mathematician?‘This book has a nice discussion of the invention vs discovery question.

His other book – “

The Golden Ratio- The Story of Phi, the World’s Most Astonishing Number” is also well worth consulting.Margaret wrote:

“mystery is a mobilizer- the ancient, the esoteric, the connections are all reasons to pursue something, whether it’s music or math. Many times I feel like I have chosen to reject the theory that seems correct in favor of the one that I love. And why not? Who says that we are not allowed to create meaning where we can? ”

= = =

When I do not find myself enteraining Didionesque doubts, I agree with this entirely.

= = =

Mark: The answer to your question is “because we are not computers.”

An earlier comment of mine has disappeared into the netherworld, leaving my comment about enchantment hanging with no attachment.

Rescued. For some reason, it was in the spam folder.

That happens quite a bit. I need to check it every day.

Mark, you asked “how can you believe something which you also believe is less likely than an alternative view to be the case? (Serious question. I am not trying to be tricksy.)”

I suspect we are on the same page here, and when I say “create meaning where we can,” I really mean that last part. I’m not trying to create meaning where science has already provided a meaning, or where a meaning otherwise exists. But in the humanities, and especially anthropology/sociology/ethnomusicology, there are so many things that we never CAN know (although we keep trying!) that the projection of meaning sometimes feels like the majority of what the field is. The question “what does music do?” is a good example of one of those- the answer, on Didionesque days (as Dan K said above!), is “it depends,” or “not much,” or “it fools people,” or “nothing.” But, since the answer is slippery and elusive, and always will be, I can push against this with my own scholarship.

Caveat: these are pretty unformed thoughts, and I’m just thinking about this for the first time now- please excuse any raw edges!

Seth wrote: “I think when we gain a certain virtuosity or fluency in some medium it becomes tedious to engage with that medium at lower levels of fluency. I know when I learned how to juggle 5 balls and became proficient at that level it became tedious. Then I adapted by learning to alter the shape of the pattern while listening to music and it became fresh an engaging again.”

I understand what you mean! For me subjectively, though, I’m not talking just about lower levels of fluency (tragically). I can certainly be amazed by the technical achievements of a professional musician- I can marvel at how they do what they do, and I can academically appreciate their hard work and/or “talent” (whatever that means), and I could think things like “that sound is beautiful.” But I’m constantly calculating in my brain: oh, she did that in that way. Oh, I don’t like that phrasing – I DO like that phrasing. Oh, that interval was very in tune. If I were playing it, I’d do it the same/differently.

We’re lucky to have the BSO and all of its fabulous musicians around here, and hearing them play is basically perfection. In other words, I very much still appreciate music, but most of the time, the magic is gone. It’s a rare performance that is magical (and I’m talking, maybe once every three or four years). I suspect that it has less to do with the presence on stage than it does with me.

Margaret –

Again I find your clarifying comment interesting. I am wondering if it is just my inferior understanding of music theory, my lack of knowing-how from a performance standpoint, and my less refined ear that allows music to captivate me on a more regular basis. Your description seems to indicate that is difficult for you to listen to the music outside of the theory laden critic orientated framework. But is not there much to a musical performance that is under-determined by theory and a critical ear. Technique is one thing, but I find myself drawn to the performers emotional investment in what they are putting out. Do they mean what they are doing – that is what draws me in (I think). Even I can hear errors of execution from time to time, but if the other element is there those errors don’t detract much from my experience.

Growing up I played basketball daily and loved the game. I would characterize many of those pickup games as among the most sacred experiences of my life. I know a lot about the game and the skills involved, and I marvel at the talents of those displayed on the highest level. Yet I would rather watch a competitive pick-up game then the NBA all-star game. The all-star game has devolved into a display of talent and technique, just the components of the game – the game itself along with it’s real beauty is absent.

I may be naive, yet I wonder if fluency with the components of craft need necessarily detract from the ability to experience it organically as a whole. I guess it depends among other things on the craft ( how completely do the components explain the whole ), and the level of fluency of the viewer.

“Mind independent” – yes, if we assume an atemporal mathematics, it would have to be mind-independent, and that makes no sense.

What we call gravity is simply a mechanistic relationship between two bodies of matter. Newton’s ‘law of gravity,’ which gives us a mathematical formula for gravity, is merely a human measure of that relationship, it is not the relationship itself.

Mathematical forms and formula can only be presumed mind-independent if they are discovered in the most brutish manner possible – by literally stumbling on a formula that a rock had somehow careved into itself. Otherwise, every form and formula is a construction by human minds.

It is not only Pragmatism that sees problems here. But I reference it because as a Pragmatist I reject the Platonic notion of truths independent of human interaction or intervention. There is certainly a world around us, and it is just as it is, but knowledge concerning it is only what we can come with in interactions with it – and which, given the adequate invention including devising new technology, we can actually change. ‘Truth’ is just what works in order to enhance our interactions and improve our interventions. Wisdom is thus not a matter of knowledge, but a matter of compassion, insight, tolerance, resilience, and willingness to adapt.

“Yes, metaphysical a prioris are what really lies at the heart of opposition to Platonism.” This is a weak remark, since Platonism is itself a metaphysical a priori.

But I don’t see how you can really get a generative mathematics or computation out of Platonism, since every mathematical innovation would have to be a discovery of something that is already held to exist, and I don’t remember personal computers existing in Plato’s day.

I’m not being churlish; my point is that the innovations we see in technology, dependent as it is on mathematical innovation, seem to me incompatible with Platonism on some level, although wholly compatible with Pragmatistic instrumentalism.

Platonism was developed with a Naive Realism that assumed the world was as it would always be, and perfect forms came to us either perfectly through our senses, or from our souls. There is a certain beauty to this; but by Aristotle in the next generation it was becoming clear this wasn’t the whole story, since it couldn’t adequately account for differences of perception, different cultural experiences, and the need to respond directly to social influences in a creative way. When we enter the Modern era, the potential for change and difference has multiplied thousands’ fold. Perfect forms – of any sort – only exist in our heads. Mathematics can measure our reality, and can be used to reconstruct our reality, but it cannot take the place of that reality.

I suspect Plato would have held non-Euclidean geometry suspect as some sort of parlor trick; and n-dimensional geometry would probably have been outlawed from the Republic, as potentially creating confusions.

I’ve said that I was always a fictionalist. For the most part, I do mathematics in about the same was as a mathematical Platonist.

I think the first time that I saw a difference, was with the Continuum Hypothesis (or CH for short). I was in graduate school, and we in a coffee room discussion of Gödel’s proof that CH is consistent with AC (axiom of choice). Later Cohen proved that CH is independent of AC. For me, as a fictionalist, that completely settled the issue. But a Platonist in that same discussion insisted that CH is either true or false in his Platonic universe, and that the proof of independence is therefore an unsatisfactory answer. I’m inclined to think that you have to get into rather esoteric areas (such as CH) before much disagreement shows up.

I disagree with your point about non-Euclidean geometry. Mathematical Platonists were just as interested in that. The resistance to non-Euclidean geometry had more to do with a belief that the cosmos was Euclidean.

Welcome CJ, it was a fun essay, though (given comments by Neil, EJ, and Seth) I don’t have much to add, except… having dealt with mathematical Platonists at other debate sites I sort of (selfishly) hope you don’t make too many converts, and perhaps come to love promoting Fictionalism?

Platonists may be having a lot of fun on the inside, but (in debate) they can be pretty exasperating on the outside 🙂

I’m still reading Jane McDonnell’s

The Pythagorean world: Why mathematics is unreasonably effective in physics. For fans out there of multiverses, she discusses Hamkins paperThe Set-theoretic Multiversehttps://arxiv.org/pdf/1108.4223.pdf

“The multiverse view is one of higher-order realism – Platonism about universes – and I defend it as a realist position asserting actual existence of the alternative set-theoretic universes into which our mathematical tools have allowed us to glimpse… In particular, I argue that the continuum hypothesis is settled on the multiverse view by our extensive knowledge about how it behaves in the multiverse…”

As to music, there is that literature on dichotic listening by musically untrained and trained individuals. One doesn’t want to go too far “left-right” brain – it is not that simple, but non-musicians tend to attend more to the right ear, while with training this shifts to the left ear. Hearing does not cross like other sensorimotor stuff, so non-musicians tend to rely on right hemisphere processing (suggested also by split brain experiments). So maybe the right hemisphere is more easily impressed 😉

Thank you all so much for all the welcoming and challenging comments. I’m very happy to be a part of this group. I wish I’d had a calmer weekend in which I could have responded in kind to what I could.

As it sits, I’m a bit overwhelmed by the sheer quantity of responses and very proud to have been a part of a debate between philosophy of mathematics that didn’t and with fuming egos and rattled friendships (as they often do in my experience).

labnut-

“But what if Platonism really was true? Mathematics would still look the same. You would still reach this conclusion but you would be wrong. So what have you shown, other than you want to believe in Fictionalism?”

I’ll admit I’m a bit shaken by this prospect. I never considered myself wanting to believe in Fictionalism, more so that I begrudgingly admitted to myself that this was the most likely true belief while simultaneously keeping a metaphoric shrine to Plato in the back of my mind. I’ll have to give some more thought to some of your comments that you have made along with this one.

Mirowley-

“But mystery is a mobilizer”

I just had to put this in my comment because it’s beautiful! Thanks for this comment.

dbholmes-

” I sort of (selfishly) hope you don’t make too many converts, and perhaps come to love promoting Fictionalism?”

I’m not sure you have to worry about that. In perspective of my dialogue, I was speaking with teenagers. Since my acceptance that Platonism is unlikely, I’ve not tried to convince anyone that it might truly exist. Rather, I find it can be a good avenue for mathematical inspiration. For example, my student ‘Klark’, a like minded person to me, would need no such inspiration from me because he sees the beauty of mathematics irrespective of his understanding of axioms and assumptions. It simply offers a more easily painted beauty for those less interested in the art of mathematics.

Finally, since I’ve seen many questions about it, I’ll try to recall the ‘prank’ in response. I say try because I am often spending my days tormenting ‘Klark’ in an endless tit-for-tat exchange spawning from his first hacking attempt on my computer. I believe this particular exchange consisted of our IT director and I convincing him that the $1200 computer science team laptop had been stolen after he absentmindedly left it unattended in a computer lab. Of course, what had really happened was the IT director found it, handed to me, I locked it away in a cabinet and we proceeded to hatch our mutual revenge. It was quite a fun 24 hours!

Thank you again to all of you for your interest and participation. I feel very welcome and am looking forward to the next discussion!

DB,

“

Platonists may be having a lot of fun on the inside, but (in debate) they can be pretty exasperating on the outside”I wish DM were here to argue his case. I often disagree with him but I enjoy his thoroughly contrarian spirit and insightful comments.

EJ,

you seem to have rather gone off course in your reply to my comment. To remind you, here is what I originally said, so let’s stick to what I said(it is always a good idea to quote your interlocutor’s words so that this problem can be avoided!):

“

But what if Platonism really was true? Mathematics would still look the same. You would still reach this conclusion but you would be wrong”This seems to have given you the impression that I am defending Platonism.

No I am not. You know my beliefs, from earlier encounters, so you have no reason to think I defend Platonism.In fact I believe that Platonism is mistaken(although I enjoy DM’s spirited defence of Platonism! I wish he was here). The problem is that we cannot distinguish between Fictionalism and Platonism merely by examining the mathematics we have, as CJ did. That quite simply was my point and that is all I was saying. Your many paragraphs have talked past what I said because you have wrongly assumed that I defend Platonism.

The properties of mathematics are suggestive of Platonism, because, if Platonism were true, we would observe these properties in mathematics, which is why the debate is possible.

To sum up, I think that Platonism is plausible but wrong. The really fundamental problem that the opponents of Platonism face is fourfold:

1) how could rational order emerge from the random chaos of nothingness?

arising from this

2) what is the origin of the inexorable, immutable, time, place and circumstance invariant laws of nature?

3) how can biological machines perceive and create rational order?

related to this

4) how could biological machines perceive and create the True, the Good and the Beautiful(the three great transcendentals)

Brandishing the buzzword, ’emergence’ does not explain anything. It is a label, a place holder, empty of content, used to disguise the fact that we don’t have a bloody clue.

Theism, Pantheism, Panentheism and Platonism address this problem by arguing that rationality is fundamental to the universe and has always existed.

Atheism claims instead that somehow(!) it all emerged(how?) from the random chaos of nothingness. This is by far the least plausible explanation. It is a belief and not an explanation coming from people who deride beliefs and pride themselves on explanations!! If you dispute this, try giving me a good explanation of how the laws of nature emerged from the random chaos of nothingness when there were no laws of nature to guide their emergence!

You can think of Platonism as pre-existing order without God. In fact the only rational stance for an atheist is to believe in Platonism. DM understood this very well and I consider him to be one of the few rational atheists, hence my great respect for him.

David,

“

I’m still reading Jane McDonnell’s The Pythagorean world: Why mathematics is unreasonably effective in physics”Thanks for that valuable reference, which is so apposite to the conversation. I lost no time getting the book and my first impression was an ‘Aha’ moment of wonder when she said(I cheated by jumping to the conclusion)

“

I ventured to introduce a broad metaphysical framework based on the principles of mind and mathematics. In order to explain how physical reality condenses out of mathematical structure, I used Leibniz’s ploy of identifying the fundamental constituents of the universe with simple minds—what he calls monads. I considered an interpretation of the set theoretic hierarchy in which the individuals are monads. This interpretation makes an analogy between all of reality—Being—and an infinite mind. Being has structure which is mathematical, due to the arrangement of parts, and it has thoughts. Its thoughts are interpreted as correlated states of monads. Being thinks about itself, its own structure, and creates physical reality through a process of self-actualisation.”“

I concluded that mathematics is about the structure of Being; that it is unchanging, necessary and true; and that we come to know about it by abstracting structure from the world around us and by self-reflection. Mathematical objects do not exist in a mysterious Platonic heaven totally independent of the physical world. Rather, physical reality is a mental interpretation of a subset of mathematical structure. Mathematics is applicable because it truly describes the fundamental structure of reality. No mathematics is surplus. I made a distinction between mathematics and human mathematics which is a cultural product of our society.”“

One of the key drivers of Pythagoreanism for physicists is their sense of the unreasonable effectiveness of mathematics in physics, so that was my starting point. It led to a picture of the world in which mathematics and physics reflect the internal structure and processes of Mind. This view is more accurately described as mathematical idealism than Pythagoreanism”From the Amazon blurb about the author:

“

Jane McDonnell is Adjunct Research Associate in the Philosophy Department at Monash University, Australia. She has doctorates in both theoretical physics and philosophy and over twenty years’ experience applying mathematics in academia and industry. She has authored or co-authored more than eighty technical papers in physics, mathematics, finance and philosophy.“Arguments foe Fictionalism seem to confuse two separate issues. If I say ‘if A then B” and somebody disagrees saying (as often happens) “But you don’t know if A is true or false”, then they are assuming that the truth or otherwise of “if A then B” depends upon the truth or otherwise of A. But of course it doesn’t.

So, of course it is only a convention that 13 follows 12. However this is irrelevant to the truth or otherwise of Platonism (and I have no opinion on whether Platonism is true or false).

A mathematical truth does not depend upon such-and-such axiom being true or false, rather it is that such-and-such facts follow from such-and-such axioms.

So if I adopt a set of axioms such that the number following 12 is 37, then it wouldn’t render the statement “12+1=13” false or uncertain it would only mean that I have added a new way of using the symbol ‘+’ to the many ways it is already used in maths. So “12+1=37” would be true for this new meaning of ‘+’.

So the argument that fictionalism must be true because the axioms are only conventions appears to make the same mistake as saying the truth or falsity of “if A then B” depends on the truth or falsity of A.

So I would like to hear an argument for fictionalism which does not depend upon the premiss that the axioms are conventions, because this premiss is true for both Fictionalism and Platonism.

I started as a Fictionalist. If you asked me to define mathematics I would have said that is easy, mathematics consists of choosing a set of symbols and then making up some rules to manipulate them by and then proceeding to manipulate them by those rules.

By and large my maths lectureres and fellow students agreed Nothing disnaying about that, the idea did not put me off mathematics.

But I couldn’t ignore the problems with the position. I think it was Lee Smolin’s attack on Platonism that turned me completely away from Fictionalism and made me see that something like Platonism was closer to the way things really are.

Smolin is forced to depend on various Platonic assumptions and is forced to invent a kind of pop-up temporary ad-hoc Platonic realm without explaining exactly what brings them into existence or why the same facts will be brought into existence by two people independently coming up with the same axiom.