Then it would be impossible to create a reasonable hierarchy like you were talking about, if the meaning of tight and loose could vary. — Metaphysician Undercover
y the way do relativity theory speak about rules about the mathematical objects used to write its laws with? Aren't those mathematical objects a part of the theory? I don't think relativity theory assumes that numbers for example have a mass, or that they move with a speed less than light, etc.. Those mathematical objects are fixed, eternal, unchangeable. It's the physical objects that the rules of relativity theory applies to. I don't think that the mathematical objects and rules that it uses has anything to do with relativity theory. Imagine that number 1 for example will rut with time? That's crazy! Isn't it. — Zuhair
We use possibly fictional objects to display the mathematical rules with, because this is the most evident way in which it can be presented. Most of these rules, as well as the objects manipulated are non-spatio-temporal. But I think we can have pseudo-spatio-temporal objects representing mathematical worlds, thus in some sense approximating the real world. But I think also that nothing of the rule physical world law about physical objects would be applicable to these realms either. — Zuhair
These objects might be completely fictional, imaginary, and not intended to represent the real world at all. — Metaphysician Undercover
Yes, I think there is an intermediate position. Mathematics is producing rule following obedient fictional objects and scenarios. However, those happen to have applications in the real world. I suspect that the matter is not accidental. There is seemingly some common grounds between imagination and the real world. Some rules about arithmetic works fine when applied to real objects, and it really succeeded in increasing our understanding of the real world around us. On the other hand obviously there are rules that are not applicable to the real world like having infinitely many numbers, etc... I think logically obedient rule following imaginative scenarios do have some common grounds with reality. — Zuhair
This is TRUE of many mathematical disciplines. For example a lot of set theory stuff is so imaginary that it might not even find any application at all. However, no one can really tell. Even imaginary numbers turned to have applications, even non-Euclidean Geometry turned to have applications. The problem is that we don't know really what our reality adheres to, or even what discourse about obviously imaginary objects could be useful in applications about the real world. — Zuhair
The problem is that if we take Quines-Putnam indispensability argument, then even those non-spatio-temporal features of mathematical object might need to be accepted as part of reality, even though not a physical concrete kind of reality, but some kind of reality there!? The mathematician usually do not bother with these philosophical ground. All of what he cares for is the analytic consequences of his assumption, which for clarity and simplicity they are usually stipulated outside of the confines of space or time or both, or within the confines of some imaginary world that has its own space and time characteristics, as well as its own part-whole relationship with respect to eternity issues in it. Most mathematicians work primarily in a Platonic world! Philosophy comes later! — Zuhair
Do you agree that this is the basis of that dualist separation between the real world and the Platonic world? The human mind apprehends the world as consisting of numerous possibilities. In order for it to understand each, every, and any possibility, the mind assigns to itself, the capacity to understand infinite possibilities. But that assignment is wrong, because the human mind is restricted by the real world, being a part of the human body, and so its capacity to understand is really restricted. So the human mind has created this dualist premise, and all these dualist principles, in an attempt to give itself the capacity to understand anything, and everything, when in reality it doesn't have that capacity. That Platonism is self-deception. It was far a good cause, but when it runs its course and we see that it is impossible for it to give us what it was designed to give us, we need to get rid of it. — Metaphysician Undercover
That Platonism is self-deception. It was far a good cause, but when it runs its course and we see that it is impossible for it to give us what it was designed to give us, we need to get rid of it. — Metaphysician Undercover
Your views here suite "Mathematics for science", while some mathematicians might insist on "Mathematics for Mathematics". — Zuhair
I agree with the duality policy. The real issue is how to judge when a mathematician is going a stray? I mean as far as possible contribution to knowledge is concerned (i.e. application). I think a real foundation of mathematics must help direct mathematicians towards producing more beneficial mathematical theories. But how to judge this? I think this is a very important question? We need a foundation for applicable mathematics! But I'm almost very sure that a lot of mathematicians, possibly the most, wouldn't care the hell for that, they'll view it as too restrictive, and favor diving deep into the world of logically obedient rule following scenarios, no matter how wildly far their imaginative worlds are from reality. Sometimes I think this is like the dualism of religion and state in secular states. Let the mathematicians dive deep into the imaginary platonic world they like, and let science work with its strict observance to reality moto. The important matter is not to confuse both. We only need to coordinate both at applications! — Zuhair
The real problem is even if it is false, still the logically obedient strict rule-following themes it negotiates can prove to be extremely useful, even if in part. The real problem is that we'll never know at which stage it will "run out its course"? Possibly one day foundations for 'applicable' mathematics would issue, having clear cut edge between what is beneficial and what is not?! Perhaps by then this platonic dream would vanish! perhaps?! but I don't really know where such a thing would start? or even if it could start really? Until such alternative is found, we'd better keep the current dualist stance. — Zuhair
The mathematician might say 'we have to be able to apply the infinite, it's part of mathematics', but really all that the infinite is, is a thing of beauty, a beauty which is negated by any misguided attempt to apply it. — Metaphysician Undercover
Can you describe this "need" for me? If mathematics prior to the 19th century got along fine without speaking about the infinite, where does this need to apply the infinite, in set theory, come from? — Metaphysician Undercover
I'd say that It is not just infinitude of natural numbers that we need for the sake of such unlimited measurement, we need to stipulate useful relations and functions (operators) on them, so relations like "equal", "smaller than", "greater than", and functions like "summation", "multiplication", "exponentiation" etc.. all of these are needed. So we need SENTENCES in a language that uses those functions and relations between natural numbers, those sentences would be the axioms, and can be the theorems deduced in the systems having logical and mathematical inference rules starting from those axioms. So I insist that we need "sentences" in the language of arithmetic, which doesn't include only natural numbers, but also includes relations and functions on those natural numbers. — Zuhair
So we need SENTENCES in a language that uses those functions and relations between natural numbers, those sentences would be the axioms, and can be the theorems deduced in the systems having logical and mathematical inference rules starting from those axioms. So I insist that we need "sentences" in the language of arithmetic, which doesn't include only natural numbers, but also includes relations and functions on those natural numbers. — Zuhair
Now due to Godel's incompleteness theorems, there is no effective system that can capture all true sentences of arithmetic! Now notice here that I'm speaking about sentences of arithmetic and their terms only range over natural numbers, i.e. there is no infinite object whatsoever symbolized in that language, so it is totally about finite objects or descriptions about finite objects, so it is the kind of language that we think it can possibly have applications in our world, viewing all objects in our world being finite. — Zuhair
Now we come to the role of theories about the infinite, i.e. theories that speak about infinite objects like actual infinite sets for example, which as you said, and I think it is generally agreed that our physical world seems to be incompatible with their existence in it. However, despite this incompatibility those infinite theories can prove some true arithmetic sentences (those that only range over natural numbers using finitely long formulas) to be true that the theories restricted to the finite objects fail to prove! — Zuhair
Now we want the infinite to give us the capacity of measurement as you said, but you need the tools for those measures, and the tools for those are not just the existence of infinitely many naturals, but we need sentences about some relations and operations on them, and the main problem is that we don't have a theory restricted to the finite realm that can effectively give us all of those sentences, which are infinite in number by the way. Or even if those useful arithmetical sentences are finite in number, still we don't have a theory about finite objects that can capture all of those finite sentences, or even if we can have, we don't know which theory is that. — Zuhair
So theories speaking about infinite objects can indeed prove some of those arithmetical sentences about the finite realm of them, and those can be useful sentences. So that's why we go to the infinite. — Zuhair
Of course there are other more radical objections to your line of view, like the mathematics for mathematics viewpoint, and like the other direction objection that is our physical world itself being of ACTUAL INFINITE reality and that our current physical theories and observations being erroneous about that aspect, etc... I didn't want to go to those, because I honestly think that the bulk of evidence supports a finite (or at most potentially infinite) outcast of our universe, and that mathematics ought to be useful in understanding that universe, and therefore I approached it from that perspective as given above. — Zuhair
Hi Zuhair, I'm having a little difficulty understanding some parts of your post, but I'll provide my interpretation with some criticism, and you can tell me where I misunderstand. — Metaphysician Undercover
but this is a self-defeating assumption because it assumes an object which is immeasurable, and the purpose of assuming the infinite is to make all things measurable — Metaphysician Undercover
The only objects that PA speaks about are naturals which are in some sense measures of finite objects. So generally speaking PA would be the kind of a theory that is expected to have applications about objects in our finite (or potentially infinite) universe. So all sentences written in the language of PA are statements about finite objects, so they all speak about the state of affairs related to finite objects, as we regard them to be potentially applicable! — Zuhair
The problem is that MOST of sentences written in the language of PA are not provable in PA. So we are missing a lot of sentences that might have useful application in our real world, because PA cannot prove them. However those arithmetical sentences can be proven from theories that encounter speech about existence of infinite objects, like set theory for example, so ZFC can prove arithmetical sentences which cannot be proven in PA. Notice that I'm speaking about arithmetical sentences, i.e. sentences about natural numbers, i.e. statements about measurement of the FINITE, so those are statements that can have applications in our real world, and some of those sentences are provable in ZFC while PA cannot prove them! — Zuhair
So you are not differentiating between the 'absolute capacity' of measurement, which is sometimes ironically called by some set theories as the absolute infinite, [which you call the "infinite" by the way], and the various grades of the infinite, the latter ones are using your terms qualities, and they can be measured in an effective manner, while the former one (the absolute infinite) is what you cannot measure nor can formalize it as an axiom, and using your terms I would describe it as not really a quality, its a pure quantity (using your terms), this absolute infinite is something that no set theorist tries to capture by its axioms or theories, its an unlimited tendency of measurements. — Zuhair
However, my argument above (the one you've answered to) is not that deep. It only says that theories that have capture SOME infinite objects, are vastly stronger (deductively speaking) than theories that only capture the finite, that's why technically speaking, those stronger theories can help even prove some theorems of strictly weaker theories that only speak about the finite, not only that it can prove theorems spoken about in their language that those weaker theories cannot prove, and those sentences are of the kind speaking about infinite objects and relations between them and properties of them, so they are (generally speaking) the kind of sentences expected to have application in our finite world. This mean that theories speaking about infinite (as well as finite) objects can aid in measurements of the actual world via proving those sentences of them that are concerned with the finite realm of them. It supply us (technically speaking) with more and more sentences about finite objects, and so enrich our knowledge base and potential to make descriptions in our finite world. Its a pure technical issue. So they are useful and can make contributions to our finite world, although they are theories that have the capability of speaking of infinite objects (pure unities with infinitely many qualities). — Zuhair
Therefore I believe that "infinite object", or "infinite objects" represents a misunderstanding of the natural boundaries which objects have. Unless "infinite object", or "infinite objects" can be given some real meaning, as referring to something real in the world, supported by real principles, it's simply nonsensical talk.. — Metaphysician Undercover
If it doesn't make sense in relation to the real world, then it cannot be a true premise. Therefore the proofs which are derived are unsound — Metaphysician Undercover
You and others in philosophy might underestimate it, because this second role is in principle dispensable! But there is a great difference between "in principle" and "in practice", I'd agree that they are in principle dispensable, but in practice they are not, because we are humans, so theorems of sound axiom systems that are provable from very long proofs will not be discovered by the human mind, while the assisting stronger systems would enable discovering those theorems because they can prove them in shorter steps, and then afterwards we can go back to the original sound theory and find the long proof of those theorems. — Zuhair
I don't understand why you call these theories, which are not based in sound premises "stronger theories". They are clearly weaker. — Metaphysician Undercover
You refer to such a theory as a "technical guide", and say that they are aimed at practice. So lets say that they are like hypotheses. We apply them in the attempt to prove whether they are true or false. So we must be willing to reject them when they are proven to be false. — Metaphysician Undercover
*stronger* is a logical term. Theory A is stronger than theory B if and only if every statement provable in B is provable in A, but not every statement provable in A is provable in B. — Zuhair
This is challenging! If it fails and proves misleading, then we REJECT the extended system from being a part of useful mathematics, and only keep it as a piece of beautiful analytic school of art (Mathematics for Mathematics). — Zuhair
The tool might be the most primitive, awkward tool, but if it brings us success in what we are doing, then we are not inclined to look for a better one, That success misleads us because it hides the fact that we really need a better tool, by making it appear like we have the tool we need — Metaphysician Undercover
However, I do think that imperfections would sooner or later show themselves, no matter how much useful they are. And at that point the habit will break. — Zuhair
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