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I believe that "platonism" in this article refers to mathematical platonism, i.e. the view that mathematical entities actually exist, as opposed to mathematical

I believe that "platonism" in this article refers to mathematical platonism, i.e. the view that mathematical entities actually exist, as opposed to mathematical formalism, i.e. the view that mathematical entities are made up. https://en.wikipedia.org/wiki/Philosophy_of_mathematics#Platonism https://en.wikipedia.org/wiki/Philosophy_of_mathematics#Plat... https://iep.utm.edu/mathplat/ https://iep.utm.edu/mathplat/ This view is only called platonism because it parallels Plato's thoughts, but it is not an explicit reference to Plato's original philosophy. Modern platonism (after Kant) exists as an attitude about a kind of non-mental realism of abstract objects, and so it is with mathematical platonism. These days, there are very few Platonists (those who subscribe to the philosophy of Plato), but many platonists. As for the second point: A PhilPapers survey showed that most professional philosophers of mathematics are platonists, and most mathematicians who are familiar with the debate lean towards platonism. It is much more intuitive to understand how mathematical claims are true if you believe that numbers are real objects in the fist place, which in turn allows us to use the same accounts of truth we would normally use for non-abstract objects. At least prima facie, mathematical truths like 2+2=4 do not seem like they are different types of truth than empirical truths like "Biden is currently president". If mathematical objects like 2 and 4 weren't real, we would have to come up with a new account of truth that applies specifically to mathematics to explain what makes its statements true, which while possible, is a lot less elegant. Obviously, that doesn't refute platonism's alternatives, but in the words of a friend, it would really suck if he spent his career studying something that was entirely fictional, so he better be a platonist.

dr_dshiv 4y ago
Plato is worth reading in the original because of the way it makes one think. But we might call it Pythagorean if we want to claim that the world is made of Math. Roger Penrose has a very nice “3 worlds theory” about mathematics, the physical world and the world of mental experience. He describes theories for how these worlds encompass each other and overlap. In “The Road to Reality.”
- techno_tsar 4y ago
  In addition to this, Plato is worth reading because he is still relevant and his work outlined the agenda of huge swathes of western philosophy, which undeniably influences our scientific paradigm and political worldviews. I have not yet read The Road to Reality. At 1094 pages, I hope it is not extremely complicated.
  - dr_dshiv 4y ago
    Plato gets summarized — and that mostly doesn’t work. Need to read it. The Parmenides is supposed to be the hardest to understand—pah, I’d start with that. Cut to Zeno arguing conclusively that all is one. He then conclusively argues that everything can’t be one. Then… Don’t be put off by the size of Penrose’s road to reality! The introduction and first chapter already made the book worth it. I’ve not gotten much further myself (but that’s how I read)
    - DiscourseFan 4y ago
      Plato can't be summarized, there are no conclusions to any of the arguments, just endless arguments. That's why he's so great, he always leaves the question open, there is always room for greater understanding.
DiscourseFan 4y ago
>Obviously, that doesn't refute platonism's alternatives, but in the words of a friend, it would really suck if he spent his career studying something that was entirely fictional, so he better be a platonist. Remember, fiction is real: it takes time to be written, its distributed to its audiences, it has a definite cultural impact and often changes how people view the world. Entirely fictional plays have spawned political revolutions--there are many forces at play in a fictional work that are certainly objective and real. There isn't anything that isn't "real", the question is not "is math real" (it definitely is), the question is "where does math exist?", and on account of its location (inside or outside the head), how we should be approaching it and where it should be applied. Nobody would say math isn't useful, but we might say it could be more useful if its basis is more rigorously examined.
GoblinSlayer 4y ago
Doesn't it mean that everything is true? Is 2+2=4 the same kind of truth as "elves live in Lorien"? Or more math: assume 0/0=1, then 0/0+0/0=1+1=2=(0+0)/0=0/0=1, therefore 2=1. It's math, so it must be truth too?
- gallopingcomp 4y ago
  2+2=4 is (roughly) the same kind of truth as “two groups each consisting of two elves have a total of four elves”, or “if you travel a distance of 2cm twice, you’ve traveled by 4cm”, except it isn’t tied to the real or fictional existence of centimeters or elves. And loosely the same kind of truth as “imagine a world where elves live in Lorien… in this world, elves live in Lorien.”) And on the second point, by assuming 0/0=1, either you have left the realm of natural numbers (or real numbers), or you have to break the distributive law of addition, or all the symbols mean completely different things. Otherwise, you are essentially declaring both 1!=2 and 1=2, which is not math.
  - GoblinSlayer 4y ago
    That's a tautology, you can similarly say "imagine a world where mathematics isn't tied to the real or fictional existence, in this world mathematics isn't tied to the real or fictional existence". >And on the second point, by assuming 0/0=1, either you have left the realm of natural numbers (or real numbers), or you have to break the distributive law of addition, or all the symbols mean completely different things. I didn't do such things. > Otherwise, you are essentially declaring both 1!=2 and 1=2, which is not math. It's derived from initial assumptions, which is how all math works.
    - gallopingcomp 4y ago
      1. I would object to the “similarly”, because they are not similar types of statements. And yes, the tautology aspect is the whole point of the axiomatic method (which has limitations that cannot be directly blamed on that premise). 2. You didn’t do the first two. But the symbols now mean different things than their conventional interpretations in number theory. 3. > It's derived from initial assumptions, which is how all math works It’s exactly how _logic_ works, and is how all math works, but that would only qualify it as (il)logic, and not inherently math. Necessary but not sufficient condition. (Sorry ;)
gallopingcomp 4y ago
As a (former) student of psychology, I personally subscribe to the view that both platonism and constructivism are true (edit: in that they both accurately depict different-but-interrelated aspects of mathematics). It’s a false dilemma much like “is light a wave or a particle” or “free will or determinism”. (Yup, I am that “you can have it all” pollyanna type of person.)
- igravious 4y ago
  as an aside I feel like you're misusing the notion of what a Pollyanna is: https://www.collinsdictionary.com/dictionary/english/pollyanna https://www.collinsdictionary.com/dictionary/english/pollyan... “a person who is constantly or excessively optimistic” To be a Pollyanna (I think) is to be Panglossian: https://www.merriam-webster.com/dictionary/Panglossian https://www.merriam-webster.com/dictionary/Panglossian “marked by the view that all is for the best in this best of possible worlds : excessively optimistic” This sort of thinking goes back to Leibniz (and probably a lot further) “We live in the best of all possible worlds” https://www.britannica.com/topic/best-of-all-possible-worlds https://www.britannica.com/topic/best-of-all-possible-worlds So to be a Pollyanna is to have a certain (overly?) (irrationally?) optimistic towards ones situation in life and perhaps even the nature of human suffering in general. To be contrasted with the Buddhist thought which asserts that basically life is suffering: https://en.wikipedia.org/wiki/Four_Noble_Truths https://en.wikipedia.org/wiki/Four_Noble_Truths === Anyway, back to your belief that (mathematical) Platonism and (mathematical Constructivism can be reconciled. So a hard-core mathematical Platonist believes – if I am not mischaracterising their position – that things like numbers are actually existing entities that we discover and that even if we humans had never existed that numbers would have or that if we humans cease existing that numbers will continue to exist. So mathematical progress is a progress of discovery, not creation. A hardcore mathematical Constructivist believes that if it were not for us numbers would not exist, the very idea of number would be unthinkable, we think numbers into being. A radical Constructivist (like me) does not believe in actually existing infinities or infinitesimals and so does not believe in actually existing unbounded real numbers like irrationals and transcendentals except in a symbolic or algorithmic sense. Can these two positions be reconciled? Can we optimistically reconcile them. I think not. But I do think that the slighter weaker proposition that certain formal entities like numbers are necessary, I'm going to say, "truths" in that the nature of the universe necessitates certain types of mathematical entities such that once there are a sufficient class of thinking things to think them they'll pop into being, so to speak. I am open to correction on any point. I know that my personal position is more-or-less anathema to most mathematicians I've had the pleasure of sharing these ideas with (as in I've gotten into heated drunken debates/arguments about this stuff).
BlueTemplar 4y ago
Yeah, this seems to be among the hardest questions that mankind has to deal with, so it's probably not surprising that these definitions are often confused : «Platonism without Plato» : https://samzdat.com/2018/01/26/platonism-without-plato/ https://samzdat.com/2018/01/26/platonism-without-plato/ (Though it's perhaps worth it to start reading from one/two blogposts before that one.)
igravious 4y ago
I've talked to a fair number of mathematicians down through the years and the vast majority I've talked to are Platonists. Some have even thought about this stuff and know what Platonism means and assert that they are Platonists. For me, once you read up on L.E.J. Brouwer's intuitionism and think through its consequences properly you can't be a Platonist, but that's just me – I undoubtedly have a prior more fundamental belief that makes me think this way. Like, I think it'd be difficult to be a (mathematical) Platonist and an atheist at one and the same time, but many mathematicians are.
mejutoco 4y ago
Another good source for Platonism in Mathematics https://plato.stanford.edu/entries/philosophy-mathematics/#Pla https://plato.stanford.edu/entries/philosophy-mathematics/#P...
voldacar 4y ago
Don't these arguments often essentially become about the definition of "real" or "exists"? It seems to me that I can make mathematical Platonism true or false by using a weaker or stronger definition of "exists"

dr_dshiv 4y ago

Plato is worth reading in the original because of the way it makes one think. But we might call it Pythagorean if we want to claim that the world is made of Math. Roger Penrose has a very nice “3 worlds theory” about mathematics, the physical world and the world of mental experience. He describes theories for how these worlds encompass each other and overlap. In “The Road to Reality.”

techno_tsar 4y ago
In addition to this, Plato is worth reading because he is still relevant and his work outlined the agenda of huge swathes of western philosophy, which undeniably influences our scientific paradigm and political worldviews. I have not yet read The Road to Reality. At 1094 pages, I hope it is not extremely complicated.
- dr_dshiv 4y ago
  Plato gets summarized — and that mostly doesn’t work. Need to read it. The Parmenides is supposed to be the hardest to understand—pah, I’d start with that. Cut to Zeno arguing conclusively that all is one. He then conclusively argues that everything can’t be one. Then… Don’t be put off by the size of Penrose’s road to reality! The introduction and first chapter already made the book worth it. I’ve not gotten much further myself (but that’s how I read)
  - DiscourseFan 4y ago
    Plato can't be summarized, there are no conclusions to any of the arguments, just endless arguments. That's why he's so great, he always leaves the question open, there is always room for greater understanding.

DiscourseFan 4y ago

>Obviously, that doesn't refute platonism's alternatives, but in the words of a friend, it would really suck if he spent his career studying something that was entirely fictional, so he better be a platonist. Remember, fiction is real: it takes time to be written, its distributed to its audiences, it has a definite cultural impact and often changes how people view the world. Entirely fictional plays have spawned political revolutions--there are many forces at play in a fictional work that are certainly objective and real. There isn't anything that isn't "real", the question is not "is math real" (it definitely is), the question is "where does math exist?", and on account of its location (inside or outside the head), how we should be approaching it and where it should be applied. Nobody would say math isn't useful, but we might say it could be more useful if its basis is more rigorously examined.

GoblinSlayer 4y ago

Doesn't it mean that everything is true? Is 2+2=4 the same kind of truth as "elves live in Lorien"? Or more math: assume 0/0=1, then 0/0+0/0=1+1=2=(0+0)/0=0/0=1, therefore 2=1. It's math, so it must be truth too?

gallopingcomp 4y ago
2+2=4 is (roughly) the same kind of truth as “two groups each consisting of two elves have a total of four elves”, or “if you travel a distance of 2cm twice, you’ve traveled by 4cm”, except it isn’t tied to the real or fictional existence of centimeters or elves. And loosely the same kind of truth as “imagine a world where elves live in Lorien… in this world, elves live in Lorien.”) And on the second point, by assuming 0/0=1, either you have left the realm of natural numbers (or real numbers), or you have to break the distributive law of addition, or all the symbols mean completely different things. Otherwise, you are essentially declaring both 1!=2 and 1=2, which is not math.
- GoblinSlayer 4y ago
  That's a tautology, you can similarly say "imagine a world where mathematics isn't tied to the real or fictional existence, in this world mathematics isn't tied to the real or fictional existence". >And on the second point, by assuming 0/0=1, either you have left the realm of natural numbers (or real numbers), or you have to break the distributive law of addition, or all the symbols mean completely different things. I didn't do such things. > Otherwise, you are essentially declaring both 1!=2 and 1=2, which is not math. It's derived from initial assumptions, which is how all math works.
  - gallopingcomp 4y ago
    1. I would object to the “similarly”, because they are not similar types of statements. And yes, the tautology aspect is the whole point of the axiomatic method (which has limitations that cannot be directly blamed on that premise). 2. You didn’t do the first two. But the symbols now mean different things than their conventional interpretations in number theory. 3. > It's derived from initial assumptions, which is how all math works It’s exactly how _logic_ works, and is how all math works, but that would only qualify it as (il)logic, and not inherently math. Necessary but not sufficient condition. (Sorry ;)

gallopingcomp 4y ago

As a (former) student of psychology, I personally subscribe to the view that both platonism and constructivism are true (edit: in that they both accurately depict different-but-interrelated aspects of mathematics). It’s a false dilemma much like “is light a wave or a particle” or “free will or determinism”. (Yup, I am that “you can have it all” pollyanna type of person.)

igravious 4y ago
as an aside I feel like you're misusing the notion of what a Pollyanna is: https://www.collinsdictionary.com/dictionary/english/pollyanna https://www.collinsdictionary.com/dictionary/english/pollyan... “a person who is constantly or excessively optimistic” To be a Pollyanna (I think) is to be Panglossian: https://www.merriam-webster.com/dictionary/Panglossian https://www.merriam-webster.com/dictionary/Panglossian “marked by the view that all is for the best in this best of possible worlds : excessively optimistic” This sort of thinking goes back to Leibniz (and probably a lot further) “We live in the best of all possible worlds” https://www.britannica.com/topic/best-of-all-possible-worlds https://www.britannica.com/topic/best-of-all-possible-worlds So to be a Pollyanna is to have a certain (overly?) (irrationally?) optimistic towards ones situation in life and perhaps even the nature of human suffering in general. To be contrasted with the Buddhist thought which asserts that basically life is suffering: https://en.wikipedia.org/wiki/Four_Noble_Truths https://en.wikipedia.org/wiki/Four_Noble_Truths === Anyway, back to your belief that (mathematical) Platonism and (mathematical Constructivism can be reconciled. So a hard-core mathematical Platonist believes – if I am not mischaracterising their position – that things like numbers are actually existing entities that we discover and that even if we humans had never existed that numbers would have or that if we humans cease existing that numbers will continue to exist. So mathematical progress is a progress of discovery, not creation. A hardcore mathematical Constructivist believes that if it were not for us numbers would not exist, the very idea of number would be unthinkable, we think numbers into being. A radical Constructivist (like me) does not believe in actually existing infinities or infinitesimals and so does not believe in actually existing unbounded real numbers like irrationals and transcendentals except in a symbolic or algorithmic sense. Can these two positions be reconciled? Can we optimistically reconcile them. I think not. But I do think that the slighter weaker proposition that certain formal entities like numbers are necessary, I'm going to say, "truths" in that the nature of the universe necessitates certain types of mathematical entities such that once there are a sufficient class of thinking things to think them they'll pop into being, so to speak. I am open to correction on any point. I know that my personal position is more-or-less anathema to most mathematicians I've had the pleasure of sharing these ideas with (as in I've gotten into heated drunken debates/arguments about this stuff).

BlueTemplar 4y ago

Yeah, this seems to be among the hardest questions that mankind has to deal with, so it's probably not surprising that these definitions are often confused : «Platonism without Plato» : https://samzdat.com/2018/01/26/platonism-without-plato/ https://samzdat.com/2018/01/26/platonism-without-plato/ (Though it's perhaps worth it to start reading from one/two blogposts before that one.)

igravious 4y ago

I've talked to a fair number of mathematicians down through the years and the vast majority I've talked to are Platonists. Some have even thought about this stuff and know what Platonism means and assert that they are Platonists. For me, once you read up on L.E.J. Brouwer's intuitionism and think through its consequences properly you can't be a Platonist, but that's just me – I undoubtedly have a prior more fundamental belief that makes me think this way. Like, I think it'd be difficult to be a (mathematical) Platonist and an atheist at one and the same time, but many mathematicians are.

mejutoco 4y ago

Another good source for Platonism in Mathematics https://plato.stanford.edu/entries/philosophy-mathematics/#Pla https://plato.stanford.edu/entries/philosophy-mathematics/#P...

voldacar 4y ago

Don't these arguments often essentially become about the definition of "real" or "exists"? It seems to me that I can make mathematical Platonism true or false by using a weaker or stronger definition of "exists"