I have a great respect for this author, but I do have to note that the debate of "constructivism" vs. "platonism" doesn't really exist in Philosophy per se, because it relies on a kind of basic misunderstanding of Plato in general. At least since Kant, the general consensus in Philosophy about Plato's "forms" or ideal objects is that they are conceptual objects, famously Kant argued that while reality in-itself is inscrutable, the means of judgement whereby we organize the world into concepts (beginning with the intuitions of time and space) are not, and by analysis of judgement we can begin to better understand the process of reasoning and therefore more rigorously understand the means by which empirical judgements are made (and thus further our understanding of empirical reality).
Of course, many have attempted to answer the question of the psychological/biological explanation for how humans have shared concepts like this and shared, intelligible language. The most prominent thinkers of the 20th century, on both "sides" if you will, were Lacan on the "continental" end, and Chomsky on the "analytic" side. (Even though most Anglo-American--i.e. Analytic--philosophers would not accept Chomsky's logic as valid, and Lacan often referenced Anglo-American philosophy in his work...the categories are basically meaningless at this point.) But there are many other names, many people asking the above question, and I will not say who's thought attracts me the most, but I will say that most people I meet who study math are not familiar with them or the intellectual trends they are a part of, and they do not have a rigorous enough understanding of what philosophy is, really, to make claims about the usefulness of philosophy for their work--or even debate it, for that matter.
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.
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.
>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.
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.)
I don't think it's a false dilemma, because the essence of mathematical platonism isn't really that forms exist, but that these forms are how the universe really works.
They're not an approximation, a metaphor, or a point of view. They're the real deal - the base mechanisms. And they can be discovered through the scientific process, with its combination of physical and speculative analysis.
Clearly this is nonsense. Math isn't truly self-consistent, physical research is limited to a range of lab-friendly experiments garnished with some astrophysical guesswork, and all of it gets filtered through consciousness, which we have no clue about.
What we have is a "looking for the key under the light" situation where can only explore the things we can see. We don't know what's in the dark, and it's actually very likely that our consciousness is extremely limited and unable to perceive essential detail.
But because (tautologically) we can't see it we just assume it's not there, and our tiny and contingent view is gloriously universal.
I find the cat metaphor very revealing. Cats share a space with us but they literally do not see the same objects we do. They perceive weight, texture, and dimensions, they're far more sensitive to smell, and they have some innate models for dynamics and mechanics.
But they have no concept of the meaning of a book, a laptop, a wifi card, a Netflix subscription, or a mathematical description of General Relativity. Unless we breed them specially for intelligence for a good few tens of millennia they never will, because cat consciousness is too small to contain those concepts.
It's ridiculously, almost comically naive to believe - purely on faith - that human consciousness isn't severely limited in some analogous ways.
We're quite good at the human equivalent of hunting for food - which includes manipulating physical materials and crude energy sources, with some meta-awareness of abstraction.
What are the odds that's all there is to understand about the universe?
> I don't think it's a false dilemma, because the essence of mathematical platonism isn't really that forms exist, but that these forms are how the universe really works.
Consider Euclidean geometry, which is fine, mathematically, but, it turns out, not how the universe works. If mathematical platonists were concerned about whether mathematics' forms are how the universe really works, surely they would insist on the verification of their axioms before proceeding? And then, would mathematical platonism not be just the uncontroversial parts of the physical sciences?
The author reprises the platonism / formalism issue in this article's final section, beginning with the paragraph "But there is still more to be said. Perhaps, after all some of those Big Picture questions do remain lurking in the mathematical background." He refers to Platonists as realists, but, I think, in the sense that the forms are real regardless of whether they are how the universe really works.
I would consider the reverse (that mathematics is constructed) as just as nonsensical. One could argue that if mathematics were constructed, we are essentially taking on faith that mathematical properties in the physical universe just so happen (by coincidence) to correlate with the mathematical principles we have invented. But this seems backwards. The Pythagorean theorem makes more sense as something we have discovered, or the inner corners of a triangle add up to 180 degrees (half a circle); alien civilizations likely have arrived at the same conclusion. The simple answer is that math is simply a feature of the universe.
Even if mathematics isn’t truly self-consistent (it is not), that does not commit one to formalism or constructivism. The belief that abstract entities, if they are real, must be self-consistent, requires us to believe that self-consistency is a precondition for the realism of abstract entities to begin with. But there is no obvious reason to believe this.
As for the limits of our human consciousness: arguably there is a “floor” where we can have strong beliefs in the hypotheses we form about the universe (including those of mathematics). In fact, the essence of Platonism (and where it derives it’s name) is the very view that abstraction is realer than concrete or empirical particulars because it is more unchanging and absolute. It seems inconceivable to find a single world where 2+2 != 4, but we can conceive of worlds where say, Biden is not currently president, or where gravity had a different strength. In other words, the laws of logic (and perhaps many parts of mathematics) seem very fixed, but our other laws less so. Plato thought this told us something about the ultimate hierarchy of metaphysics; modern mathematical platonists like Godel think that we have a mathematical intuition that allows us to perceive mathematical objects; mathematicists like Max Tegmark thought that nothing other than mathematical objects exist at all.
It is this intuition towards the abstract as real, realer, or realist that motivates platonism. To committed platonists, the burden of proof is actually on the non-believers, partly as a preservation of logic and mathematics. If we dismiss that (a common logic), we might be incapable of having a real discussion in the first place. Whether or not that intuition is enough, or free of problems (it is not) is very debatable. However, platonism is not trivially or obviously false.
> If mathematics were constructed, we are essentially taking on faith that mathematical properties in the physical universe just so happen (by coincidence) to correlate with the mathematical principles we have invented.
Err, no.
The mathematics we as human have constructed is a superset of the physical universe - the mathematical world embraces much more than the mere confines of the physical world we can kick and observe across.
Indeed we have constructed various mathematic worlds that are at odds with each other - some my have application in this physical universe which then precludes others from also corresponding to the same.
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
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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).
Wow, I guess my mental model (so to speak) is even more “radical” than yours. I don’t think mathematics is really part of the (empirical) universe, but that they are their own kind of abstract entity. They may happen to correspond to certain patterns in how things exist and interact in the “real world”, or to sentient beings’ reasoning and modeling faculties, but they are not tied to the real world either way.
For comparison: To me, for a number to exist in a “symbolic or algorithm sense” is to for it to exist, period - but in the sense of “creating”/“discovering” a new number system to contain them. The set of rational numbers isn’t really “special” to me. (Non-negative natural numbers are “special” for their association with cardinality, but I will refrain from going down that rabbit hole this time.) (I assume you meant “unbounded” in terms of expansion into elementary algebra; do correct me if I misunderstood you.)
(i.e. Existence=NaN because it’s a loaded word, Abstractness=Yes, Independence=I have some but limited sympathy for the neo-Fregean view on this, “creating” and “discovering” are the same thing to me)
(Which would make me a platonist to some people, an intuitionist to others, I guess)
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.”
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.
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)
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.
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.
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.
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.
> or all the symbols mean completely different things
It's actually not entirely unproductive to consider this line of thought, whereby in this formulation equality actually means something like "arrivable via some number of zero divisions". I'm sure you could find all sorts of curiosities with this mathematical "toy".
Yup. You run into that all the time in abstract algebra. Although people usually don’t like to touch the equality sign; the usual practice (based on my limited exposure) is to invent equivalent operator notations.
Exactly - it’s still there (and still has its own semantics), but it’s also inert.
And, to address your point directly, of course mathematics detached from context can be used in practice. One can certainly create-slash-discover an abstract algebra, derives theorems about it detached from outside context, and then later on discover a context in which the abstract structure is applicable, and apply the pre-derived theorems.
I actually agree with you - “the symbols mean something different now” isn’t a bug, it’s a feature. But I was trying to point out (what I saw as) a big ambiguity in parent’s comment.
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.
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 :
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"
Being stirred by your last sentence, I have a lot of trouble reading some philosophy. I have read your comment several times, and I am still not sure what it is saying.
> I do have to note that the debate of "constructivism" vs. "platonism" doesn't really exist in Philosophy per se, because it relies on a kind of basic misunderstanding of Plato in general.
What is the misunderstanding, exactly? Did you end up addressing this? The next sentence is a long run-on, and I'm getting lost in what you're saying Plato actually said versus what you think people think he said. And I think I agree with another commenter that you are confusing Platonic and platonic.
And what does philosophy say about constructivism? If it doesn't address it versus some notion of idealized objects existing, then why does that matter to mathematics? It is a legitimate line of thinking and isn't an exclusively philosophical subject.
> Of course, many have attempted to answer the question of the psychological/biological explanation for how humans have shared concepts like this and shared, intelligible language. The most prominent thinkers of the 20th century, on both "sides" if you will, were Lacan on the "continental" end, and Chomsky on the "analytic" side. (Even though most Anglo-American--i.e. Analytic--philosophers would not accept Chomsky's logic as valid, and Lacan often referenced Anglo-American philosophy in his work...the categories are basically meaningless at this point.)
There are just names and labels here. What are you saying here?
It's very weird that you would reach for Lacan and Chomsky. Neither of them are really seen as credible in the field of philosophy, nor have they had really large impacts in philosophy. Cultural criticism and linguistics? Sure, I guess. But not in philosophy. Even in the 20th century "continental" tradition of cultural criticism, Lacan is not very high in the eyes of many. Heidegger (controversially, of course) or Maurice Merleau-Ponty or Paul Ricoeur have been much more influential. For whatever reason, Lacanian analysis did burn bright in the 70s and 80s, but man has it fallen hard.
Arguably all modern philosophy of language in the 20th century was influenced by Ludwig Wittgenstein. Wittgenstein is, for many philosophers, the most influential philosopher of the 20the century. Wittgenstein's "method" and work in Philosophical Investigations also offers the most far reaching and consequential criticisms of platonism, cartesian mind-body dualism, and philosophical skepticism, in the philosophical canon.
I do agree with your closing comments though. Most people aren't very well versed in philosophy and operate on a sort of pop understanding of it. They simply lack the framework to make credible claims on the usefulness of it in general. It's especially hilarious when they try to say one of the longest and most fruitful intellectual disciplines in human civilization is "not very useful." Sure, bud.
> It's especially hilarious when they try to say one of the longest and most fruitful intellectual disciplines in human civilization is "not very useful." Sure, bud.
One of the longest? Sure. One of the most fruitful? I'm not so sure. What are the concrete, usable conclusions of philosophy, and what are the fruit of those conclusions? How do they compare to the fruit of the usable conclusions of Newtonian mechanics or Maxwell's Electrodynamics?
Unfortunately much of philosophy is very easily ignored. The philosopher will get very defensive about this, as perhaps they should, since their work was the "initial" work leading to the development of a rigorous field (physics, mathematics, psychology, etc). However, much of philosophy is intentionally vague because the philosopher is grasping at straws, is speculating.
Philosophy is a discipline whose success is desperately trying to reclassify itself from its own subject. In that regard, when you say that mathematicians do not care and cannot argue philosophy, you are merely being defensive. There is nothing in the philosophy of math that a Mathematician could not grapple with at your level of expertise or better in a simple conversation with no preparation. The issue is that what you have to say is not rigorous, it is not grounded in anything, it is vague. This is your problem, not the mathematicians.
> Kantianism is not the predominant view among working metaphysicians today
Which gladly nobody claimed to be the case. "At least since Kant" gives us a time frame not a causal link — and even if it did pretend there was a causal link, any philosopher wouldn't have to be a Kantian in order to agree to a thing Kant said about Plato's concept. Or phrased differently: A Non-Kantian can agree with Kant on a thing Kant said about Plato without suddenly becoming a Kantian.
What the poster said here is to my knowledge correct (I studied philosophy).
Of course, many have attempted to answer the question of the psychological/biological explanation for how humans have shared concepts like this and shared, intelligible language. The most prominent thinkers of the 20th century, on both "sides" if you will, were Lacan on the "continental" end, and Chomsky on the "analytic" side. (Even though most Anglo-American--i.e. Analytic--philosophers would not accept Chomsky's logic as valid, and Lacan often referenced Anglo-American philosophy in his work...the categories are basically meaningless at this point.) But there are many other names, many people asking the above question, and I will not say who's thought attracts me the most, but I will say that most people I meet who study math are not familiar with them or the intellectual trends they are a part of, and they do not have a rigorous enough understanding of what philosophy is, really, to make claims about the usefulness of philosophy for their work--or even debate it, for that matter.