https://en.wikipedia.org/wiki/Philosophy_of_mathematics#Plat...
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.
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.”
I have not yet read The Road to Reality. At 1094 pages, I hope it is not extremely complicated.
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)
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.
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?
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.
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.
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".
>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.
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 ;)
Mandatory gesturing towards Gödel.
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.)
To be a Pollyanna (I think) is to be 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
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
===
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).
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)
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?
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."
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.
These are literal examples of things proven to be true constructions as a result of Axioms (of a Euclidean Geometry).
> The simple answer is that math is simply a feature of the universe.
If (big if) in fact the physical universe is a Euclidean universe (and not, say, just a tiny bit hyperbolic [1])
[1] https://www.maa.org/book/export/html/115525
> 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.
«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.)
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.
https://plato.stanford.edu/entries/philosophy-mathematics/#P...
...you just made this up. Kantianism is not the predominant view among working metaphysicians today. It's a large plurality, at best.
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).
> 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?
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.
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?
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.
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.