A mathematical theorem is a case of an abstract object. Some philosophers hold that these are real, along with perhaps propositions, but deny the reality of universals in a broader sense. To get at the broader sense with a question, are our mental general categories of “dog,” “tree,” “human,” “star,” “flower,” “goodness,” “tennis raquette,” and so on really existing in objective reality in a mind-independent way? Or are they merely human invention and convention that exist only in the mind (or not even at all)?
Note, I’m not referring to the English word “dog” (or any other English word), but to the concept that English word refers, and which may be referred to by a different sound/lettering/symbol/gesture in a different language.
The theorem itself is an abstract object just like the word “dog”.
But what the object describes is extremely concrete and exists outside the minds of men. If a scrap from a landslide breaks out as right triangle, then the square of the hypotenuse is equal to the sum of the square of the other sides.
This seemingly universal truth exists whether we know it or not.
As to other abstractions like beauty and god that have no materially observable object, they would most certainly be particulars. Objects that exist only to the observer, even if others claim to share the observers opinion.
I guess I’ve just been steeped in empiricism far too much to appreciate the complexity you apparently see.
While some of the philosophers from Greece’s golden age might have tried to ply the idea to the abstractions of their day like “essences”, it’s important to realise that some of the ideas of these undeniably brilliant men have indeed ended up in the trash heaps of philosophy.
Or at least so I’ve been taught at a public university in 100 and 200 level courses. Ergo I don’t have a doctorate on the subject so I’ve still plenty to learn.
It looks like that in order to discuss the subject in the way you fellows wish to discuss it, I have to at least tacitly agree on a few semantics that I’m unable to.
More seriously, it looks like he did not do a very good job. Someone who gives an introduction to Philosophy should not present just the opinion that he himself holds.
Are you certain about that…?
Now, while such a claim is vulnerable to retortion (trying to apply the principle to itself), in this case there is one more difficulty for you.
That is, there are facts which we do know with complete certainty. Which facts? Let’s see:
That is what has to happen, if your solution to the problem of universals is correct.
If you are right, mathematicians must keep measuring various triangles, looking for exceptions to various theorems.
So, have you ever seen mathematicians at work?
No, they are not busy measuring triangles.
Because they have a proof. And because of that they have complete certainty. If measurements show otherwise, at best Euclidean geometry did not apply there. At worst (and most commonly) that is a result of an experimental error.
So, let’s return for the “second round” of retortion.
You made a claim that happens to be a universal proposition. And according to it, you have to look for exceptions for every universal proposition. So, did you look for an exception for it? Let me guess: no. Did your professor look for exceptions to it? You can be pretty certain that no, he did not.
So, how did you accept it? Because of a sophistical, fake proof. Precisely the way that this claim was supposed to rule out.
But Gettier problem does not even try to show that “true certainty is practically impossible”. It shows that defining knowledge as “justified true belief” leads to difficulties in cases when one gets “lucky”.
Lol, no. But as a coincidence, he was another Scotsman.
As he would certainly also say. But this touches on the “trash heap of philosophy” that I mentioned.
This is not attempt to anger anyone, but in a world and educational setting where empiricism rules the day it can be rather expected. These views are what ultimately separates western philosophies from eastern. And by eastern, I don’t mean Byzantine. I mean more eastern than that.
There is no perfect certainty. Only functional certainty we employ to live our lives.
Like Gettier knew his sheep were really sheep?
No, at that point “the gallery”, as my old prof would call it, has had centuries. Moreover, there’s a possible Nobel in store for folks who overturn long-standing laws. So there’s more than no incentive for minds better than mine to do it.
For better or worse, in order that we don’t have to retread literally all human knowledge with every single individual (as literally no one has sufficient lifespan to do), there is a bit of a tacit agreement that when experts present the greatest and best truths of their field, they’re probably not trying to trick us.
The whole, over-arching point to Gettier’s problem is that it’s impossible to be completely sure that you “know”. Applied, it then shows that “believing” and “knowing” are different points on the continuum of certainty and that if a man who deals in sheep can be wrong about a sheep, then it’s possible, however infinitesimally, that we can be wrong on any claim.
The modern sciences are rather built on this proposition. No claim is ever shut to analysis. For example, gravity was a “pull” when I was a kid. Now, there are no “pulls”. Gravity is a push.
I can elaborate if needed.
Fun discussion, I appreciate it and look forward to your next reply.
So, let’s look at Gettier problem with sheep (https://www.iep.utm.edu/gettier/#H4). In it one looks at the field, sees something that looks like a sheep, and believes that there is a sheep on the field. In fact, there is a sheep in the field, but the sheep-like thing that is seen is really a dog disguised as a sheep. So, we have a belief that is true and justified, but hard to call knowledge.
Certainty does not play any role in this example (after all, the belief is still right).
And even if it did, at most that would give one (1) hypothetical case where we shouldn’t have had perfect certainty. That only shows that we do not always have perfect certainty, and:
That is what just about everyone agrees with anyway; see, for example, “theological notes” (“de fide” indicates stronger certainty than “sententia certa” or “possibile”).
That is not anywhere close to “true certainty is practically impossible”.
Trying to get “true certainty is practically impossible” from it is precisely what your empiricism is ruling out (you are not looking for exceptions, but hastily overgeneralising from a single case).
OK, so, that would explain why mathematicians are not busy measuring specifically triangles. But it still does not explain why mathematicians are trying to construct proofs instead of just trying out many examples. For example, P=NP is considered an unsolved problem, but according to your philosophy, it shouldn’t be (after all, smart people were not able to construct a good algorithm for NP-complete problems for a long time, which would indicate that none exists).
For example, what if an orbit of a planet would be found to be “wrong”? Your philosophy suggests that in such case one would have to abandon laws describing gravity. But when that actually happened, Le Verrier actually made a contrary assumption: that the laws are right, but there is a new planet. And that’s how Neptune was discovered.
So, the result is that your philosophy:
Does not match how mathematicians do research.
Does not match how natural scientists do research.
Does not match how you came to believe it.
That’s a lot of “exceptions”…
Now the thing your philosophy itself advises in such case is to abandon it.
That’s all I’m trying to say, with the minor correction that Gettier wasn’t talking about sheep. They were merely representative. He was talking about virtually any knowledge claim.
I wouldn’t make any such claim as “it shouldn’t be”. At this point, I recall Dr. H pointing out that the shoddiest rhetoricians are those that need to put words in other peoples mouths to make their point.
If you need my conclusion on something or you’d like me to apply my proposition to it, feel free to ask!
To the point though, at that point I would make the suggestion that perhaps it can’t be solved although a Nobel may be in it for the mathematician that manages to do so. Nothing more.
Absolutely so. The notion that we can be wrong is why we can revise ideas, no matter how fundamental they may be. Again, when I was a kid gravity was a pull generated by massive bodies. Everyone “knew” that. Now, gravity is a push generated by the space-time massive bodies displace - like how the water “pushes” against your boat to keep it afloat.
And, hold on to your armchair, it may yet be revised again. And again. And again. Because we never really “know”. We only “know to the best of our ability”. Gettier.
My philosophy suggests that either another factor creating the anomaly in the orbit of that planet would exist or yes indeed - we would need to revisit the theories of gravity.
At this point, your conclusion would be based on a very unsound premise, making it not worth much.
In our other discussion I mentioned I was working on an essay about this topic. Hopefully I’ll actually complete it, then I’ll share.
If you never heard the term before, I might start by saying that abstract objects are truths, and the question is whether these are real truths in the natural world, real but only of the mental representations we have in our head, or merely conventional and axiomatic. The number one (1) is an abstract object. I’m not referring to the written symbol, or the language we use to refer to it, but to the actual concept itself. 1+1=2 is another abstract object. “Snow is white” is also an abstract object. Again, let me reiterate I’m not speaking of language but of the concept or notion being referred to by the language. The language is conventional, of course. The question is whether the concepts being referred to are real in nature, real only in the mental representations we have of them, or always conventional. If the general concepts are not real in nature, or real in our minds, and everything is particular, then isn’t what I mean by 1+1=2 not the same as what you mean as 1+1=2? The claim of those who deny any reality to them is that every instance is particular and there is nothing really common between them, so it seems an issue. Some go further than just mathematical abstract objects or propositions, though, and also include all general categories (“universals”). Dog, wolf, giraffe, human, golf club, rose bush. Are these categories real such that all particulars in that category really belong to that category? Is it not real in nature but there is a real commonality between the mental representations in our head? Are they not real at all? Again, we seem to run into a similar issue if there’s no reality at all, because then what are general categories at all? How do we use them? How do we communicate them if even the instances you have in your mind and I have in my mind are particular with no real commonality between them? Maybe as a materialist you’ll brush off notions of “dog” and “tree.” Those are conventional to certain arrangements of fundamental particles, maybe? But what about at the fundamental level? Electrons and muons and up quarks and down quarks? Are these real categories? Mental? All particulars with nothing really common between them? But if there’s no commonality how do you and I possibly have any common knowledge and understanding? If there’s nothing actually common between this electron and that electron and general categories are just conventional, then how do we apply that to real science? Because a study of one particular that has nothing in common with another particular shouldn’t tell us anything or give us any predictive ability.
[Continued in next post because my rambling exceeded the character limit]
Excuse the train of thought. I’m just rushing it all out right now. Hopefully I’ve conveyed the idea. I’m not asking for a response here. There are developed nominalist schools of philosophy that attempt to answer these questions on what general concepts are, the applicability of science, etc… I think they have issues, but the point is that I’m not just making these conflicts up. There are real philosophical objections/seeming conflicts that nominalists have worked to explain and address in order to justify a nominalist approach.
I didn’t really go into the dispute between those who think the general categories are real in nature vs. those who think they’re real but only between the mental representations in our head, those two approaches are called realism and conceptualism (or idealism), respectively. As noted, the position that general categories or common notions are not real either in nature or in the mind (and that there are only particulars) is called nominalism.
Is the proposition (undyed) “snow is white” true in all languages? Is it still true even if no one’s around to think it? Is there any real commonality between the thought I have in my head when I think it and the thought in your head when you think it?
Regarding something more complicated like War and Peace, is there a truth in nature that the arrangement and relationship of concepts referred to within was possible, and that this possible arrangement is always real?
I’d use a stronger term like “discovered” for strict truth claims, however. Something like mathematics. There is a sense, as you put it, to it being used for War and Peace, but it’s looser.
We each may have objections of course. As noted, nominalism typically has to defend how it’s possible for humans to refer to general categories, the power of language, and the power of science. And those are defenses many nominalist philosophers attempt to make.
These would be your liberal use of statements such “using your philosophy” and so on.
In better environs, you simply don’t get to do that. If you want to apply your opponent’s paradigm to to an objection, you raise the objection then have them do it. Doing otherwise is almost certain to yield fallacy, strawman being probably the most common.
But to the matter at hand after reviewing my old notes, the fundamental problem is that Dr. H taught universals in a post-empiricism context.
The concessions you require to discuss it in the same way as the classics and medieval philosophers concerning constituent sub-topics like “essences” has been largely discarded and replaced with semantics.
We no longer attempt to meaningfully discuss “the essence of a dog”. We develop a semantic to define what a dog is to general satisfaction and then we proceed from there.
Essences, surely, would be an example of what Dr. H referred to as outmoded philosophy. “The trash heap” as it were.
As such, apologies for jumping in. As I suspected, this specific discussion requires concessions that most (certainly not all) which dabble in philosophy are hesitant to make. If course, this doesn’tmean the discussion is worthless. It only means that any progress made is of limited benefit and interest. Which is most certainly fine.
Apologies for typos. On phone, away from home and office. Thanks again.
Two points on this one. If all participants in this scenario were of human intelligence, they could come to understand the concept even if they were previously unfamiliar with them. You and I would agree humans don’t have infused knowledge. We come to knowledge through the senses in our encounters with them, and in some cases these could be logically explained to get the gist. If you never encountered them there may be some concepts you don’t yet have. That they don’t have the concept yet is not a refutation. The second point is that being incapable of grasping a concept doesn’t mean there is no such general category.
Is it possible for you and me to have a common understanding of all of this type without a typographical error and all of that type with the same typographical error? Manuscript A and Manuscript B? Yes? Then yes.
You illustrate my point for me. They arrive at the same concept despite having different languages, despite their particular words being different, and particular mental images being different. Either their concepts agree and have something in common, or they don’t and their understandings have nothing in common. Also, my original point in the quoted sentence wasn’t even about people speaking different languages but even just two people speaking the same language. Lastly, we’re probably going to disagree with this, but knowledge of a concept is different than a mental image. Knowledge often involves a generalization of all types in a category. The imagination and ability to generate images (or sounds or smells or tactile feels or tastes) is always particular. You and I may each never be able to develop an image in our heads of a million sided polygon. That doesn’t mean we don’t know what such things are.
These would be me ridiculing your approach to your position as “proprietary”, “closed source”, which only you are allowed to use.
And by “strawman” you obviously mean “making your philosophy look foolish”?
Yes, we know such tricks.
So, no. Such “closed source philosophy” is a great occasion to add an inconsistent “epicycle” to avoid having to avoid having to admit one’s philosophy is an inconsistent mess. And to remove it, when not needed. And to shout “Strawman!”, when someone points out the inconsistency.
Furthermore, it makes it easier to let you “develop feelings” for your philosophy, where you will feel that having to admit it was wrong would feel like a “loss of face”. It is bad enough even with “open source philosophy”, why make it still worse?
It might work, if you were known to be perfectly honest. But you are not known to be perfectly honest. As you might suspect, I do not trust you all that much (nor would I recommend you to trust me).
Well, if we can just throw various philosophies to that “trash heap” with no justification, we can also throw philosophy of “Dr. H” there too.
And if not, go and find that justification.
Yes, that’s not a work for a mere day. After all, that “throwing out to the trash heap” was an excuse not to have to study the alternative philosophies, so you do not know much about them.
That is, this “we” avoids the word “essence” and denies existence of essences, but otherwise acts as if essences existed.
We sometimes hear of savants with this ability, and their memory recall or something along those lines is fantastic. Still, I’m not speaking of constructing an imaginary image. Thomists distinguish between perceptual images, imagination, and memory on the one hand with intellection on the other, with the first three often aiding the ability of the latter. I’d be interested in reading more if you have any articles, but from what I’ve already read I don’t see a conflict.
That comment was for the exchange between Hume and MPat. To your point, I don’t know if I want to deep dive into the notion of essence here, but the terms essences and universals are not interchangeable, and on top of that, how a word is used colloquially may not fully capture its use in a technical field. I can certainly use the word “essence” in a nominal fashion to question what the heart of something is like “playing ball”, but in Aristotelianism or Thomism its technical usage is more precisely defined.