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Meh. No mention of Pythagoras or any Platonism. Max Tegmark, Karl Popper and Roger Penrose are the three best known for promoting the Pythagorean-Platonic idea

Meh. No mention of Pythagoras or any Platonism. Max Tegmark, Karl Popper and Roger Penrose are the three best known for promoting the Pythagorean-Platonic idea that mathematics precedes matter. Because that seriously freaks some people out—they can’t even deal with the idea. But, fairly basic, that triangles are transcendent and would exist in any civilization, in any galaxy? Matter has never produced a perfect sphere, but spheres are nevertheless truly real—- right?

Koshkin 4y ago
Indeed! Abstractions deal with commonalities, and those do exist. ("There exists something in common between the three horses and the three apples you want to treat the horses with.")
- colechristensen 4y ago
  So much of it comes down to what exactly you mean by exist. Which ultimately ends up being a boring disagreement. People can have bigger or smaller definitions of what it means and then have passionate disagreements with each other about what fits inside which are ultimately about nothing but how big a person prefers their definition.
  - Koshkin 4y ago
    Sure, there are many ways in which something may not exist; but the existence usually demonstrates itself in a pretty straightforward way, like with the horse who bites you if you show her that you have no more apples left.
    - ogogmad 4y ago
      Look up intuitionistic logic and constructivism to see why existence can have a weaker or stronger meaning depending on the foundations of mathematics.
    - colechristensen 4y ago
      Then you're painting yourself as one who has a certain definition of exist and can't imagine other definitions who would participate in such discussions unwittingly about differences in definition rather than the subject matter.
      - Koshkin 4y ago
        But it is existence that is the subject matter.
        jhanschoo 4y ago
        > but the existence usually demonstrates itself in a pretty straightforward way, like with the horse who bites you if you show her that you have no more apples left. I'm not sure what you are trying to say, but existence has many liminal cases. There's a rich tradition of ontologists trying to get at a good definition of existence.
- jonnybgood 4y ago
  As another comment said. What do you mean by exist? It requires a little more rigour here. Using your example: What does it mean for three horses and three apples to exist? The better question: What does it mean for three (of anything) to exist? Why not one thing, one thing, one thing? The definition of existence you appear to be using is the physical proximity of those objects. But even that can get quite hairy. If there is two apples within inches of each other and another apple 100 feet away, are there three apples or two? The answer to this question depends on what you mean by the existence of three apples.
  - akomtu 4y ago
    If an apple is cut in half, does the apple still exist? When the halves are brought together, at what distance between the two does the apple start existing?
    - ffhhj 4y ago
      Tarski would like to cut that apple into thin slices: https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox
    - anthk 4y ago
      Is there an atomic bond? No. Then, there's no whole apple any more.
  - danielam 4y ago
    > What does it mean for three horses and three apples to exist? You tell us. You seem perfectly comfortable speaking of a multiplicity of instances. (Personally, I would begin the discussion with substantial form. I can speak of many triangles that instantiate the same triangularity which allows me to assert the same properties of triangularity of them all. And yet triangularity is not to be identified with any particular triangle.)
    - Koshkin 4y ago
      This is a valuable point. Our thinking is too much tied to the language. Of course when we say that the abstract triangle exists we mean that what exists is the property of triangularity. Properties (of things) do exist, whether we like it or not. Just like the number 3, say, also exists - in the form of “threeness” property (of a collection of objects). There is not much more to be said here, except maybe that most, if not all, properties manifest themselves in, or indeed are, relations between things, and also that they are not arbitrary products of the mind, being impressed into us by the objective reality - and how can something be impressed into us that is not itself part of reality, i.e. does not exist in the most obvious sense of the word.
- Barrin92 4y ago
  the notion of commonality exists in the head of the observer because it's a useful fiction. (even 'the horse' itself is). But that is a subjective form of existence, what Jakob von Uexküll called one's "Umwelt" (the world as it presents itself to you). There's no reason to believe it precedes matter, or that triangles exist in a world without anyone to conceive of them. edit: there's a great talk by Daniel Dennett on the topic that luckily still is up on archive.org https://web.archive.org/web/20161125135644/https://www.youtube.com/watch?v=GcVKxeKFCHE https://web.archive.org/web/20161125135644/https://www.youtu...
- TuringTest 4y ago
  > Abstractions deal with commonalities, and those do exist. Mandatory reading: https://www.smbc-comics.com/comic/nouning https://www.smbc-comics.com/comic/nouning And this one is even better: https://www.smbc-comics.com/comic/real-2 https://www.smbc-comics.com/comic/real-2
sbdaman 4y ago
Edit: nevermind, this was overly critical.
- goatlover 4y ago
  Platonism is a major school of thought in the philosophy of mathematics.
  - virissimo 4y ago
    Not only that, but most philosophers who specialize in the philosophy of mathematics are Platonists: https://philpapers.org/surveys/results.pl?affil=Target+faculty&areas0=47&areas_max=1&grain=coarse https://philpapers.org/surveys/results.pl?affil=Target+facul.... IME, this is also true of working mathematicians, but much less true of physicists.
    - jhanschoo 4y ago
      What does being Platonist mean in this case; I wonder. The definition I've been given once is just that mathematical objects exist, but philosophers take it to mean that one believes in Plato's Theory of Ideas. The Theory of Ideas is a major development by Plato, but it has fatal flaws as set out in Plato's Parmenides. If so, I wonder how these Platonists have chosen to resolve it.
      - hither_shores 4y ago
        > philosophers take it to mean that one believes in Plato's Theory of Ideas. This is not accurate. Platonism, in a contemporary context, is the view that abstract objects exist. And so in particular to be platonist about mathematics is to believe in the existence of mathematical entities.
        jhanschoo 4y ago
        Thanks for correcting me.
EpiMath 4y ago
good comment. This seems to come up more in number theory than in foundations/philosophy of mathematics, but I agree is has an important place. Not just triangles, but natural numbers having a fundamental place in reality ( e.g. integral numbers of dimensions, degrees of equations at the foundations of physics, etc., etc. Daniel Shanks has a list of about 60 of these "arguments" for Pythagorean interpretation of numbers )
anthk 4y ago
Triangles' concept yes, in any of them, as long as you have two dimensions.
- tomrod 4y ago
  You can have triangles in higher dimensions, embedded in manifolds or subspaces. You can have super pathological triangles in 1 and 0 dimensions. Those aren't terribly interesting though.
steppi 4y ago
There’s an entire chapter devoted to Plato in Stewart Shapiro’s Thinking About Mathematics, the first book the author recommends. I think it’s pretty reasonable to recommend people start with an accessible contemporary survey rather than diving directly into Plato’s dialogues or any other particular primary source. As far as I’m aware, Pythagoras left no written works and would thus be unlikely to appear on a reading list.
- dr_dshiv 4y ago
  Thanks for sharing! Now I want to read it. “…would thus be unlikely to appear on a reading list.” Yes, sorry to be facetious about Pythagoras but those early ideas are still rather vibrant and they do get under people’s skin. I didn’t realize until recently what a huge effect Pythagoras had (or his transmitted ideas) on the scientific revolution, vis a vis Kepler, Galileo, Newton, Hooke, etc.
- Ligma123 4y ago
  > I think it’s pretty reasonable to recommend people start with an accessible contemporary survey rather than diving directly into Plato’s dialogues or any other particular primary source. That is basically what you should never do, in any topic, unless it's some esoteric topic and a prior top-level explanation would help. It's not like Plato is Quantum Physics.
  - adrianN 4y ago
    Reading a recent survey is the perfect thing to do if you want to get an overview of the field. There is far more primary material than one can consume in reasonable time.
    - archi42 4y ago
      I can only second this. Especially regarding the 1st Shapiro book: We had an advanced seminar that was essentially "read and discuss the essays in the book". We were about 6 people, including the (excellent) professor. It was very productive and we got a very good overview of the field back then, and I think everyone enjoyed that seminar very much. To relax from the reading, and since most (all?) of us did already have some advanced math education beyond the average philosophy student (e.g. I did major in CS, others had maths major or minor), we did a "guided by the professor" proof for the incompleteness theorem - he always wanted to do that, but couldn't in his normal lectures. That was fun. I'd suspect I wouldn't have had such fond memories if we everyone was to read the original sources, especially since that would have been nearly impossible in the given time frame. In most seminars that's usually accelerated by having one person read one source each and do a presentation and/or write an essay. I think reading a well written essay and discussing that in a small group is the superior approach. Of course if you want to become a professional in the field this will not replace reading some/most/all of the original sources in the long term. It is then a matter of opinion if it's better to start reading the source with or without some broad-but-shallow knowledge of the field. I prefer with, since I never read such material to "just learn what X said on Y so I pass some exam", but because I like thinking about it and put it into perspective [e.g. temporal, but also alternate ways of looking at things]; but then I'm not a professional in that field, so insert shrug-pony here.
silent_cal 4y ago
Transcendent triangles... hm...
criddell 4y ago
> Matter has never produced a perfect sphere, but spheres are nevertheless truly real—- right? But doesn't Tegmark say matter is mathematics? Or is that your point?
- chestervonwinch 4y ago
  parent is saying mathematics is the abstract base class. they're in the source, but you never see instances of them at runtime.
User23 4y ago
It’s also a consequence of Christian theology, namely Christ’s role as the eternal Logos. And yes of course Christian philosophy owes a huge debt to pre-Christian philosophers, particularly Aristotle. Dualism vs monism is probably the fundamental philosophical disagreement. Really though the continuing lack of respect for C.S. Peirce’s contributions to modern logic are the most galling to me. His semiotic is so full of promise.

Koshkin 4y ago

Indeed! Abstractions deal with commonalities, and those do exist. ("There exists something in common between the three horses and the three apples you want to treat the horses with.")

colechristensen 4y ago
So much of it comes down to what exactly you mean by exist. Which ultimately ends up being a boring disagreement. People can have bigger or smaller definitions of what it means and then have passionate disagreements with each other about what fits inside which are ultimately about nothing but how big a person prefers their definition.
- Koshkin 4y ago
  Sure, there are many ways in which something may not exist; but the existence usually demonstrates itself in a pretty straightforward way, like with the horse who bites you if you show her that you have no more apples left.
  - ogogmad 4y ago
    Look up intuitionistic logic and constructivism to see why existence can have a weaker or stronger meaning depending on the foundations of mathematics.
  - colechristensen 4y ago
    Then you're painting yourself as one who has a certain definition of exist and can't imagine other definitions who would participate in such discussions unwittingly about differences in definition rather than the subject matter.
    - Koshkin 4y ago
      But it is existence that is the subject matter.
      - jhanschoo 4y ago
        > but the existence usually demonstrates itself in a pretty straightforward way, like with the horse who bites you if you show her that you have no more apples left. I'm not sure what you are trying to say, but existence has many liminal cases. There's a rich tradition of ontologists trying to get at a good definition of existence.
jonnybgood 4y ago
As another comment said. What do you mean by exist? It requires a little more rigour here. Using your example: What does it mean for three horses and three apples to exist? The better question: What does it mean for three (of anything) to exist? Why not one thing, one thing, one thing? The definition of existence you appear to be using is the physical proximity of those objects. But even that can get quite hairy. If there is two apples within inches of each other and another apple 100 feet away, are there three apples or two? The answer to this question depends on what you mean by the existence of three apples.
- akomtu 4y ago
  If an apple is cut in half, does the apple still exist? When the halves are brought together, at what distance between the two does the apple start existing?
  - ffhhj 4y ago
    Tarski would like to cut that apple into thin slices: https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox
  - anthk 4y ago
    Is there an atomic bond? No. Then, there's no whole apple any more.
- danielam 4y ago
  > What does it mean for three horses and three apples to exist? You tell us. You seem perfectly comfortable speaking of a multiplicity of instances. (Personally, I would begin the discussion with substantial form. I can speak of many triangles that instantiate the same triangularity which allows me to assert the same properties of triangularity of them all. And yet triangularity is not to be identified with any particular triangle.)
  - Koshkin 4y ago
    This is a valuable point. Our thinking is too much tied to the language. Of course when we say that the abstract triangle exists we mean that what exists is the property of triangularity. Properties (of things) do exist, whether we like it or not. Just like the number 3, say, also exists - in the form of “threeness” property (of a collection of objects). There is not much more to be said here, except maybe that most, if not all, properties manifest themselves in, or indeed are, relations between things, and also that they are not arbitrary products of the mind, being impressed into us by the objective reality - and how can something be impressed into us that is not itself part of reality, i.e. does not exist in the most obvious sense of the word.
Barrin92 4y ago
the notion of commonality exists in the head of the observer because it's a useful fiction. (even 'the horse' itself is). But that is a subjective form of existence, what Jakob von Uexküll called one's "Umwelt" (the world as it presents itself to you). There's no reason to believe it precedes matter, or that triangles exist in a world without anyone to conceive of them. edit: there's a great talk by Daniel Dennett on the topic that luckily still is up on archive.org https://web.archive.org/web/20161125135644/https://www.youtube.com/watch?v=GcVKxeKFCHE https://web.archive.org/web/20161125135644/https://www.youtu...
TuringTest 4y ago
> Abstractions deal with commonalities, and those do exist. Mandatory reading: https://www.smbc-comics.com/comic/nouning https://www.smbc-comics.com/comic/nouning And this one is even better: https://www.smbc-comics.com/comic/real-2 https://www.smbc-comics.com/comic/real-2

sbdaman 4y ago

Edit: nevermind, this was overly critical.

goatlover 4y ago
Platonism is a major school of thought in the philosophy of mathematics.
- virissimo 4y ago
  Not only that, but most philosophers who specialize in the philosophy of mathematics are Platonists: https://philpapers.org/surveys/results.pl?affil=Target+faculty&areas0=47&areas_max=1&grain=coarse https://philpapers.org/surveys/results.pl?affil=Target+facul.... IME, this is also true of working mathematicians, but much less true of physicists.
  - jhanschoo 4y ago
    What does being Platonist mean in this case; I wonder. The definition I've been given once is just that mathematical objects exist, but philosophers take it to mean that one believes in Plato's Theory of Ideas. The Theory of Ideas is a major development by Plato, but it has fatal flaws as set out in Plato's Parmenides. If so, I wonder how these Platonists have chosen to resolve it.
    - hither_shores 4y ago
      > philosophers take it to mean that one believes in Plato's Theory of Ideas. This is not accurate. Platonism, in a contemporary context, is the view that abstract objects exist. And so in particular to be platonist about mathematics is to believe in the existence of mathematical entities.
      - jhanschoo 4y ago
        Thanks for correcting me.

EpiMath 4y ago

good comment. This seems to come up more in number theory than in foundations/philosophy of mathematics, but I agree is has an important place. Not just triangles, but natural numbers having a fundamental place in reality ( e.g. integral numbers of dimensions, degrees of equations at the foundations of physics, etc., etc. Daniel Shanks has a list of about 60 of these "arguments" for Pythagorean interpretation of numbers )

anthk 4y ago

Triangles' concept yes, in any of them, as long as you have two dimensions.

tomrod 4y ago
You can have triangles in higher dimensions, embedded in manifolds or subspaces. You can have super pathological triangles in 1 and 0 dimensions. Those aren't terribly interesting though.

steppi 4y ago

There’s an entire chapter devoted to Plato in Stewart Shapiro’s Thinking About Mathematics, the first book the author recommends. I think it’s pretty reasonable to recommend people start with an accessible contemporary survey rather than diving directly into Plato’s dialogues or any other particular primary source. As far as I’m aware, Pythagoras left no written works and would thus be unlikely to appear on a reading list.

dr_dshiv 4y ago
Thanks for sharing! Now I want to read it. “…would thus be unlikely to appear on a reading list.” Yes, sorry to be facetious about Pythagoras but those early ideas are still rather vibrant and they do get under people’s skin. I didn’t realize until recently what a huge effect Pythagoras had (or his transmitted ideas) on the scientific revolution, vis a vis Kepler, Galileo, Newton, Hooke, etc.
Ligma123 4y ago
> I think it’s pretty reasonable to recommend people start with an accessible contemporary survey rather than diving directly into Plato’s dialogues or any other particular primary source. That is basically what you should never do, in any topic, unless it's some esoteric topic and a prior top-level explanation would help. It's not like Plato is Quantum Physics.
- adrianN 4y ago
  Reading a recent survey is the perfect thing to do if you want to get an overview of the field. There is far more primary material than one can consume in reasonable time.
  - archi42 4y ago
    I can only second this. Especially regarding the 1st Shapiro book: We had an advanced seminar that was essentially "read and discuss the essays in the book". We were about 6 people, including the (excellent) professor. It was very productive and we got a very good overview of the field back then, and I think everyone enjoyed that seminar very much. To relax from the reading, and since most (all?) of us did already have some advanced math education beyond the average philosophy student (e.g. I did major in CS, others had maths major or minor), we did a "guided by the professor" proof for the incompleteness theorem - he always wanted to do that, but couldn't in his normal lectures. That was fun. I'd suspect I wouldn't have had such fond memories if we everyone was to read the original sources, especially since that would have been nearly impossible in the given time frame. In most seminars that's usually accelerated by having one person read one source each and do a presentation and/or write an essay. I think reading a well written essay and discussing that in a small group is the superior approach. Of course if you want to become a professional in the field this will not replace reading some/most/all of the original sources in the long term. It is then a matter of opinion if it's better to start reading the source with or without some broad-but-shallow knowledge of the field. I prefer with, since I never read such material to "just learn what X said on Y so I pass some exam", but because I like thinking about it and put it into perspective [e.g. temporal, but also alternate ways of looking at things]; but then I'm not a professional in that field, so insert shrug-pony here.

silent_cal 4y ago

Transcendent triangles... hm...

criddell 4y ago

> Matter has never produced a perfect sphere, but spheres are nevertheless truly real—- right? But doesn't Tegmark say matter is mathematics? Or is that your point?

chestervonwinch 4y ago
parent is saying mathematics is the abstract base class. they're in the source, but you never see instances of them at runtime.

User23 4y ago

It’s also a consequence of Christian theology, namely Christ’s role as the eternal Logos. And yes of course Christian philosophy owes a huge debt to pre-Christian philosophers, particularly Aristotle. Dualism vs monism is probably the fundamental philosophical disagreement. Really though the continuing lack of respect for C.S. Peirce’s contributions to modern logic are the most galling to me. His semiotic is so full of promise.