No, it is the study of being, which is not necessarily synonymous with existence. For example, one view is that ontological existence (i.e., actuality) is a subset of reality (which also encompasses some possibilities and some necessities), which is a subset of being (which also encompasses fictions).Ontology is the study of existence. Isn't it? — Metaphysician Undercover
By defining "existence" in another context-specific way, obviously. There are plenty of other terms that mean something different in mathematics than in metaphysics or in other sciences.How could there be a form of existence which isn't ontological existence? — Metaphysician Undercover
That is not just pragmatism, it is the scientific method. How else would you propose that we evaluate our hypotheses to ascertain whether they accurately represent reality?Again, we encounter the problem of pragmatism. If prediction is all that is required, then we gear our epistemology toward giving us just that, predictability. — Metaphysician Undercover
As I have explained to you several times now, no one except a platonist would claim that mathematical existence conforms to "the rigorous philosophical definition" of (ontological) existence. Everyone else understands this, so please stop belaboring your terminological objection.If you are handing to "existence" a definition which allows that an imaginary, fictional thing, exists, then it's not the rigorous philosophical definition which I am used to. — Metaphysician Undercover
That is a misinterpretation, and you know it by now.I interpret this as your "epistemic stance" requires Platonic realism as a support, a foundation. — Metaphysician Undercover
Notice that the problem is with the conception itself, it has nothing to do with "the real". The idea that we can conceive a point anywhere is false, as demonstrated by the square root problem. Conceiving of continuity in this way, such that it allows us to put a point anywhere is self-defeating. Therefore we need to change our concept of the continuity of "space". — Metaphysician Undercover
Now. let's add time to the mix. We already have a faulty conception of space assumed as continuous in a strange way which allows us to create irrational figures. Special relativity allows us to break up time, and represent it as discontinuous, layering the discontinuous thing, time, on top of the continuous, space. Doesn't this seem backward to you? Time is what we experience as continuous, an object has temporal continuity, while space is discontinuous, broken up by the variety of different objects. — Metaphysician Undercover
Your analogy is faulty, because what you have presented is incidents of something representing what is meant by the symbol "5". So what you have done is replaced the numeral "5" with all sorts of other things which might have the same meaning as that symbol, but you do not really get to the meaning of that symbol, which is what we call "the number 5". The point being, that for simplicity sake, we say that the symbol "5" represents the number 5. But this is only supported by Platonic realism. If we accept that Platonic realism is an over simplification, and that the symbol "5" doesn't really represent a Platonic object called "five", we see instead, that the symbol "5" has meaning. Then we can look closely at all the different things, in all those different contexts, which you said could replace the symbol "5", and see that those different things have differences of meaning, dependent on the context. Furthermore, we can also learn that even the symbol "5" has differences of meaning dependent on the context, different systems for example. Then the whole concept of "a number" falls apart as a faulty concept, irrational and illogical. That's why you can easily say, anything can be a number, because there is no logical concept of what a number is. — Metaphysician Undercover
Prediction is not a good indicator of understanding. Remember, Thales predicted a solar eclipse without an understanding of the solar system. All that is required for prediction is an underlying continuity, and perhaps some basic math. I can predict that the sun will rise tomorrow morning without even any mathematics, so the math is not even prerequisite, it just adds complexity, and the "wow' factor to the mathemagician's prediction. So, continuity and induction is all that is required for prediction. Mathematics facilitates the induction, but it doesn't deal with the continuity. Real understanding is produced from analyzing the continuity. This is an activity based in description, and as I mentioned, is beyond the scope of mathematics.
Again, we encounter the problem of pragmatism. If prediction is all that is required, then we gear our epistemology toward giving us just that, predictability. If this is easiest done using false premises like Platonic realism, then so be it. But we do this at the expense of a real understanding. — Metaphysician Undercover
I do not see how this can be. The constructive real line is not Cauchy-complete. It's only countably infinite. It does not contain any of the noncomputable numbers. It can not possibly be an intuitively satisfying model of a continuum. I'm troubled by this and I'm troubled that the constructivists never seem to be troubled. — fishfry
[continuation of the previous post]
So, you can see constructivist logic as an algebra of propositions built with computable functions (https://en.wikipedia.org/wiki/Heyting_algebra). You cannot build non-computable functions using only the operations of this algebra, but you can add elements that are not part of the algebra ("external" non-computable functions) to obtain a new algebra that uses all computable functions plus the function that you just added. — Mephist
That's exactly the same thing that you do adding square root of 2 to the rationals: you obtain a new closed field that contains all the rational numbers plus all that can be obtained by combining the rational numbers with the new element by using the operations defined on rational numbers. — Mephist
But I see that the main problem for you is not about the soundness of logic, but about the cardinality of the set of real numbers. — Mephist
So, my question is: how do you know that the cardinality of the set of real numbers is uncountable? — Mephist
Well, Cantor's diagonal argument is still valid in constructivist logic: it says that any function that takes as an argument an integer and returns a function from integers to integers cannot be surjective (cannot generate all the functions). The proof is exactly the same: take F(1,1) and change it into F(1,1) + 1, then take F(2,2) and change it into F(2,2) + 1, etc... This is a computable function if all the F(n,n) were computable, ( n -> F(n,n) - very simple algorithm to implement ) but it cannot be in the list: if it were in the list ( call it X(n,n) ), let "m" be the position of X ( X is the m-th function ). What's the value of X(m,m) ?
X(m,m) cannot be computable. — Mephist
The problem is well known: you cannot enumerate all computable functions because there is no way to decide if a given generic algorithm stops.
So, computable functions are as uncountable as real numbers are. Where's the difference? — Mephist
No, because even the motion of the quarks inside the proton is quantized, at the same way as the motion of the electrons is. If the proton is in it's base state (and that's always the case, if you are not talking about high-energy nuclear collisions), ALL that happens inside of it is described ONLY by an eigenfunction of the Hamiltonian operator with the lowest eigenvalue: it's a well-defined mathematical object. And all protons in their base state are described by the same function. No other information is required to describe COMPLETELY it's state (even if quarks were made of "strings" and strings were made of "who knows what"). What would change in case quarks were made of strings is that the Hamiltonian operator would have a different form, probably EXTREMELY complex, but the wave-function would be the same for all protons anyway. — Mephist
No, there isn't an exact length that can be measured with infinite precision. But you don't need to be able to measure an atom with infinite precision to check if two atoms are exactly identical: identical particles in QM have a very special behavior: the wave-function of a system composed of two identical particles is symmetric (if they are bosons) or anti-symmetric (if they are fermions) (https://en.wikipedia.org/wiki/Identical_particles). Because of this fact, the experimental result conducted with two identical particles is usually dramatically different from their behavior even if they differ from an apparently irrelevant detail.
For example you can take a look at this: https://arxiv.org/abs/1706.04231 — Mephist
Yes, I agree. There are no exact measurements, — Mephist
but there are exact predictions in QM. — Mephist
For example, the shapes of hydrogen atom's orbitals are regular mathematical functions that you can compute with arbitrary precision: http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydwf.html
Of course, you cannot verify the theory with arbitrary precision, but the theory can produce results with arbitrary precision (at least in this case). — Mephist
Yes, that's because physical experiments are more and more difficult to realize if you want more and more precision, and in this case ( the exact measurement of electron's magnetic moment ) — Mephist
even the computational complexity of the theoretical computation grows exponentially with the precision of the result. But in theory the result can be calculated with arbitrary precision. — Mephist
Yes, exactly. That's what I mean. — Mephist
Well, let's abandon the discussion about this "theory" ... :smile: — Mephist
P.S. In my opinion, that's one of the most interesting aspects of QM: the information required to describe exactly an atom is limited: it's a list of quantum numbers, each of them is an integer in a limited range. So, just a bunch of bits.
If the equations of QM were the same as in classical mechanics (orbits of planets depending on their initial conditions without any limit to the precision of measurement), chemistry would be a complete mess: every atom would be different from all the others (as every planetary system is different from all the others) — Mephist
, it is the study of being, which is not necessarily synonymous with existence. For example, one view is that ontological existence (i.e., actuality) is a subset of reality (which also encompasses some possibilities and some necessities), which is a subset of being (which also encompasses fictions). — aletheist
By defining "existence" in another context-specific way, obviously. There are plenty of other terms that mean something different in mathematics than in metaphysics or in other sciences. — aletheist
That is not just pragmatism, it is the scientific method. How else would you propose that we evaluate our hypotheses to ascertain whether they accurately represent reality? — aletheist
As I have explained to you several times now, no one except a platonist would claim that mathematical existence conforms to "the rigorous philosophical definition" of (ontological) existence. Everyone else understands this, so please stop belaboring your terminological objection. — aletheist
That is a misinterpretation, and you know it by now. — aletheist
Yes, but the problem is that (for example) particles are always detected as little spots (such as a point on a photographic plate) and wave functions are spread all over the space, or on a space much larger than the observed spot. Nobody has never seen an elementary particle that looks like a wave function! — Mephist
Special relativity allows us to represent time as discontinuous?? Why? — Mephist
On the contrary, in special (and general) relativity space and time have to be "of the same kind", because you can transform the one into the other with a geometrical "rotation" ( https://en.wikipedia.org/wiki/Lorentz_transformation ) simply changing the point of view of the observer. — Mephist
Do you agree that the volume is an attribute of an object? — Mephist
I know, this explanation is not very "philosophical"... and, to say the truth, I don't really understand why is this such a philosophical problem :yikes: But what's the problem with this interpretation? — Mephist
But I am afraid that's all what physics (at least contemporary physics) does: prediction. Nothing else! — Mephist
Maybe that is a problem, but it is a problem of physics since the beginning: Newton didn't know how to make sense of a "force" that acts from thousands of kilometers of distance. — Mephist
The reality is that there are equations that work, and you can apply a mathematical theory made of imaginary things with imaginary rules that happen to give the right results. The real "ontological" reason why this system is able to "emulate" the experiments of the real world, nobody is able to explain. And it's not only about the use the square root of 2. — Mephist
I believe you do not have a very thorough education in philosophy, or you would not characterize "abstraction" in this way. — Metaphysician Undercover
Abstraction is a process. That process is sometimes described as producing a thing which might be called "a concept", or "an abstraction". There might be a further process of manipulating that thing called "an abstraction", but notice the separation between the process which is abstraction, creating the immaterial thing called an abstraction, and the process which is fixing a name to the supposed "immaterial thing" (an abstraction) and manipulating it. — Metaphysician Undercover
To begin with, we need to analyze that process of abstraction, and justify the claim that an immaterial object is produced from this process. If there is no immaterial object produced, then the name which is supposedly given to an immaterial object, simply has meaning, and there is nothing being manipulated except meaning. But if you are manipulating meaning you stand open to the charge of creating ambiguity and equivocation. This is why we separate logic, which is manipulating symbols, from the process of abstraction which is giving meaning to those symbols. So it is very good to uphold this principle. In logic we manipulate symbols, we do not manipulate "something immaterial" (meaning) which the symbols represent. What the symbols represent is determined by the premises. The "something immaterial" (meaning) precedes the logic as premises, and extensions to this, as new understanding, may be produced from the logical conclusions, but what is manipulated is the symbols, not the immaterial thing (meaning). — Metaphysician Undercover
I don't say that I've found a "loophole", I say that there is weakness. And, it's not me who found this weakness, which is a deficiency, it's been known about for ages. You look at this deficiency as if it is a loophole, and insist that the loophole has been satisfactorily covered up. But covering a loophole is not satisfactory to me, I think that the law which has that deficiency, that weakness, must be changed so that the loophole no longer exists. — Metaphysician Undercover
Until you provide me with a definition of "field" for this premise, your efforts are futile. — Metaphysician Undercover
If a field requires set theory, I'll reject it for the same reason I rejected your other demonstration. — Metaphysician Undercover
If you can construct a field with square root two, without set theory, then I'm ready for your demonstration. If you produce it I'll make the effort to try and understand, — Metaphysician Undercover
because I already believe that you would need to smuggle in some other invalid action, because that's what's occurred in all your other attempts. — Metaphysician Undercover
You never explained to me what you mean by "mathematical existence" that remains an undefined expression. — Metaphysician Undercover
It's not the case that I have a block in dealing with symbology, but what I need is to know what the symbol represents. — Metaphysician Undercover
Until it is explained to me what the symbol represents I will not follow the process which that symbol is involved in. — Metaphysician Undercover
I believe that whatever it is that is represented by the symbol, places restrictions on the logical processes which the symbol might be involved in. Supposedly, you could have a symbol which represents nothing (though I consider this contradiction, as a symbol must represent something to be a symbol), and that symbol might be involved in absolutely any logical process. However, once the symbol is given meaning, the logical processes which it might be involved in are limited. So if you start with the premise that a symbol might represent nothing, I'll reject your argument as contradictory. — Metaphysician Undercover
"Fictional existence" is contradiction plain and simple. To be fictional is to be imaginary, and to exist is to be a part of a reality independent of the imagination. If you are handing to "existence" a definition which allows that an imaginary, fictional thing, exists, then it's not the rigorous philosophical definition which I am used to. I think that if you cross this line, you have put yourself onto a very slippery slope, denying the principles whereby we distinguish truth from falsity. — Metaphysician Undercover
Wow, this keeps getting more and more ridiculous. No one is claiming that mathematical existence has anything to do with "existing substance." In mathematics--again, except for platonism--the term "existence" does not imply anything ontological whatsoever.OK, I'll assume for the sake of argument that there is a type of existence, "mathematical existence", which is a different type of existence from "ontological existence". I'll assume two different types of existing substance, like substance dualism. — Metaphysician Undercover
Something exists mathematically if it is logically possible in accordance with an established set of definitions and axioms. The natural numbers, integers, rational numbers, real numbers (including the square root of two), and complex numbers all exist mathematically, in this context-specific sense.How would I define "mathematical existence"? — Metaphysician Undercover
Nonsense, prediction is just as much a significant aspect of the scientific method as observation. Why do we have theories? How do we come up with them? Our observations prompt us to formulate hypotheses that would explain them; this is retroduction (sometimes called abduction). We make predictions of what else we would observe, if those hypotheses were correct; this is deduction. We then conduct experiments to determine whether our predictions are corroborated or falsified; this is induction.A significant aspect of the "scientific method" involves "observation", and observation is meant to be objective. The goal of "prediction" introduces a bias into observation. — Metaphysician Undercover
One more time: No one is claiming otherwise.... I've come to the conclusion that abstractions are not existent objects. — Metaphysician Undercover
I suggest, based on our conversation, that you may be highly expert on what the great philosophers say about abstraction; but your actual experience and knowledge about how abstraction works is virtually nil. At least when it comes to mathematical abstractions. And what's more abstract than mathematical abstractions? — fishfry
In college you look at derivatives and integrals, more abstractions. — fishfry
I see no reason to abandon my casual definition, paired with my experience of grappling with mathematical abstractions. — fishfry
Aren't you conflating book learnin' with actual experience? How can you tell a math person they don't know abstraction? That's like telling a pizza chef he doesn't know marinara sauce. — fishfry
If you think the mathematical existence of the square root of two is a "weakness" or defect in mathematics, it is because you are so ignorant of mathematics, that you haven't got enough good data to reason soundly about mathematics. I would think someone in your position would be desirous of expanding their mathematical understanding. Think of it as "opposition research." Learn more so you can find more sophisticated ways to poke holes. — fishfry
Well, a field is typically defined as a type of set; but the definition really has nothing to do with set theory. It's about what algebraic operations are allowed. — fishfry
If you will grant me the existence of the rational numbers; I'll build you a square root of 2. — fishfry
Symbols don't necessarily need to represent anything. If I have a symbol that behaves a certain way; that's just as good as a thing that behaves that way. At some level one can take the symbol for the thing.
That's abstraction. — fishfry
Why, pray tell, may I not type that symbol on the page? And say that it stands for a green thing? Why can't I do that? It's the foundation of civilization. — fishfry
So I ask you? Is Ahab the captain of the Pequod? Or its cabin boy? Do you really claim to be unable to answer on the grounds that Ahab's a fictional character? Nihilism. — fishfry
2) In order to do (1) I require only one premise from you. You must grant me the existence, in whatever way you define it, of the rational numbers. If you'll do that I'll whip up sqrt 2 in no time flat, no set theory needed, and no cheating on that point. I will use no set theoretic principles. — fishfry
3) We have a sticking point, which is that you don't accept a symbol unless it comes with a meaning attached. If you truly believe that then you can't solve the snaggle problem, which requires you to reason logically about symbols whose meaning is not defined. — fishfry
4) We have stumbled on an interesting point. You say that a symbol can never be conflated with the thing it's supposed to symbolize. But in math, we often do exactly that. We don't know "really" what the number 2 is. Instead, we write down the rules for syntactically manipulating a collection of symbols; we use those rules to artificially construct a symbol that acts like the number 2. Then we just use that as a proxy for the number 2. — fishfry
This is the modern viewpoint of math. We don't care what numbers are; as long as we have a symbol system that behaves exactly as numbers should. — fishfry
In order to figure out things that we don't understand, we give them names. We write down the properties we want the names to have. We apply logical and mathematical reasoning to the names and the properties to learn more about the things. That's how science works. That's how everything works. — fishfry
So we are ALWAYS writing down and using symbols without much if any understanding of the things we're representing. It's exactly through the process of reasoning about the symbols and properties that we LEARN about the things we're interested in. That's Newton writing that force = mass times acceleration. At the time nobody knew what force, mass, and acceleration were. Newton defined those things, which may or may not "really" exist; then he applied mathematical reasoning to his made-up symbols and terms; and he thereby learned how the universe works to a fine degree of approximation. — fishfry
You reject all of this. If Newton had said F = ma to you, you'd have said, what's force? And Newton would describe it to you, and you'd say, well that's not real, it's only a symbol. You reject all science, all human progress, rationality itself. If I tell you all x's are y's and all y's are z's, and you REFUSE TO CONCLUDE that all x's are z's because I haven't told you what x, y, and z are, you are an absolute nihilist. You believe in nothing that can help you get out of the cave of your mind. — fishfry
Wow, this keeps getting more and more ridiculous. No one is claiming that mathematical existence has anything to do with "existing substance." In mathematics--again, except for platonism--the term "existence" does not imply anything ontological whatsoever. — aletheist
Something exists mathematically if it is logically possible in accordance with an established set of definitions and axioms. The natural numbers, integers, rational numbers, real numbers (including the square root of two), and complex numbers all exist mathematically, in this context-specific sense. — aletheist
Nonsense, prediction is just as much a significant aspect of the scientific method as observation. — aletheist
One more time: No one is claiming otherwise. — aletheist
Perhaps in metaphysics/ontology, but definitely not in mathematics.Any claim of "existence" is validated (substantiated) with substance. — Metaphysician Undercover
I have already done so, repeatedly.If you think that there is a type of existence which is not substantial then please explain. — Metaphysician Undercover
Perhaps in metaphysics/ontology, but definitely not in mathematics.Do you not recognize that "possible" refers to what may or may not be, so it is contradictory to say that possible things are existing things. — Metaphysician Undercover
I never claimed that it is. Prediction enables us to evaluate whether our hypotheses hold up to further experimental and observational scrutiny. The goal is knowledge, which consists of beliefs (i.e., habits) that would never be confounded by subsequent experience.What I deny is that prediction is the goal of the scientific method. — Metaphysician Undercover
You keep imposing your peculiar metaphysical terminology, as if everyone else is obliged to conform to it regardless of the context. In this case, you seem to be insisting that only an ontological existent can be the object of a symbol. In mathematics, and even in ordinary language, an abstraction can also be the object of a symbol, as long as the universe of discourse is established. The objects of the names "Pequod" and "Ahab" are a boat and its captain in the fictional world of Melville's novel. The object of the word "unicorn" is a horse-like animal with one horn; the fact that no such animal exists in the ontological sense does not preclude the word from having an object at all.It's easy to assert "no one is claiming that an abstraction is an existent object", yet everyone backs up set theory which clearly assumes that the abstraction which a symbol represents, is an object. — Metaphysician Undercover
Seriously? Due to this behavior of yours, I can find nothing else to say other than you are boldly ignorant (of mathematics and its terminology) and stubborn (about your rigid definitions).Due to this behaviour of yours, I can find nothing else to say other than you are boldly lying. — Metaphysician Undercover
Algebra makes the same mistake as set theory, assuming that a symbol represents an object. — Metaphysician Undercover
How is algebra faulty? That is, algebra is a tool. Far as I know, and in my limited experience, it is a tool that works and does and accomplishes its proper tasks. But you say no. Make your case.Go ahead, but no algebra or other faulty premises. — Metaphysician Undercover
I am trying to honestly understand, but why do you propose that sets should only include apriori existing entities, and not ones defined by the processes of inference and computation themselves. That is - logic is an algorithm and our application of that algorithm manifests the imperatives in the axiomatic system. The algorithm is inaccurate in almost all practical cases, and therefore is not exactly representative of apriori existing objects.But if we define "symbol" in this way, then we cannot use algebra or set theory, which require that a symbol represents something. — Metaphysician Undercover
Let me try. Suppose that people have to compute the ratio between the lengths of the sides and the diagonal of an object that approximates a square, but lives in some unknown tessellation of space. You are not informed of the structure of the tessellation apriori and you know that the effort for its complete description before computation is prohibitive. You do however understand that the tessellation is vaguely uniform in size and has no preferred "orientation" or repeating patterns. It is random in some sense, except for the grain uniformity. The effective lengths for the purpose of the computation are the number of cells/regions that the respective segment divides. You also know that the length will be required within precision coarser then the grain of the tessellation/partitioning itself. With this information in mind, you want practical algorithm for the computation of the ratio between the sides and diagonal of a square, to an unknown precision, which is greater then the grain of the tessellation.The real question though is whether the "object" supposedly represented by √2 is a valid object. If you assume as a premise, that every symbol represents an object, then of course it is. But then that premise must be demonstrated as sound. — Metaphysician Undercover
Perhaps in metaphysics/ontology, but definitely not in mathematics. — aletheist
You keep imposing your peculiar metaphysical terminology, as if everyone else is obliged to conform to it regardless of the context. In this case, you seem to be insisting that only an ontological existent can be the object of a symbol. In mathematics, and even in ordinary language, an abstraction can also be the object of a symbol, as long as the universe of discourse is established. — aletheist
Alright man. It's not set theory you object to, it's 10th grade algebra. It's not abstraction you object to, it's the very concept of using the symbol '2'. — fishfry
How is algebra faulty? — tim wood
Far as I know, and in my limited experience, it is a tool that works and does and accomplishes its proper tasks. But you say no. Make your case. — tim wood
I'm not looking for arcane nonsense. The sense I am interested in is analogous to your saying that knives don't cut. I have knives and used as knives, they cut. Algebra, used as algebra, "cuts." So in implying that algebra is faulty, in what sense of its proper use, when used properly, does it not "cut"? — tim wood
I am trying to honestly understand, but why do you propose that sets should only include apriori existing entities, and not ones defined by the processes of inference and computation themselves. That is - logic is an algorithm and our application of that algorithm manifests the imperatives in the axiomatic system. The algorithm is inaccurate in almost all practical cases, and therefore is not exactly representative of apriori existing objects. — simeonz
Indeed, that would be mathematical platonism, as I have acknowledged. However, I am not a mathematical platonist--I have quite explicitly denied that the symbols represent existent objects in the ontological sense.Yes, I agree, in mathematics some people make the unsubstantiated claim that the symbols represent existent objects. This is called Platonic realism — Metaphysician Undercover
Platonism is by no means the only philosophy of mathematics that employs the well-established term "existence" when referring to abstract objects. As I have clearly and repeatedly stated, for those of us who are not mathematical platonists, ontology has nothing whatsoever to do with the "existence" of such objects.You seem to believe that there is some other form of ontology, some other universe of discourse, which allows that abstractions have "mathematical existence", as objects, which is not Platonism. — Metaphysician Undercover
All you have done is obtusely stuck to your rigid terminology, refusing to pay any heed to the multiple explanations that I and others have offered to correct your evidently willful misunderstanding. I see no point in wasting my time any further.All you have done is stated Platonist principles and lied in asserting that no one is assuming Platonism. — Metaphysician Undercover
I want to know what the symbols are being used for. If you assert that the symbol "2" represents an object, I want a clear description of that object, so that I can recognize it when I apprehend it, and use the symbol correctly. — Metaphysician Undercover
Platonism is by no means the only philosophy of mathematics that employs the well-established term "existence" when referring to abstract objects. As I have clearly and repeatedly stated, for those of us who are not mathematical platonists, ontology has nothing whatsoever to do with the "existence" of such objects. — aletheist
You reject science. In science we DON'T know what something is, so we give it a symbolic name, write down the symbol's properties, and reason about it in order to learn about nature. — fishfry
When Newton wrote F=maF=ma those were made up terms. Nobody knew (or knows!) exactly what force or mass is. Acceleration's not hard to define. But even then Newton had to invent calculus to define acceleration as the second derivative of the position function.
You reject all that.
Nihilism. — fishfry
Again, your peculiar metaphysical terminology is not binding on the rest of us.But "object" refers to a very specific type of thing, a unique individual, a particular, having an identity as described by the law of identity. — Metaphysician Undercover
Apparently not--an object is whatever a logical subject denotes, which can be an abstraction or a concrete existent.That I reject the notion that properties which are described by concepts like "force" "mass" and "acceleration" are themselves objects, doesn't make me nihilist. It just means that I understand the difference between an object and a logical subject. — Metaphysician Undercover
Again, your peculiar metaphysical terminology is not binding on the rest of us. — aletheist
Apparently not--an object is whatever a logical subject denotes, which can be an abstraction or a concrete existent. — aletheist
Cantor's theorem. |X|<|P(X)||X|<|P(X)|. This is a theorem of ZF, so it applies even in a countable model of the reals. You mentioned Skolem the other day so maybe that's what you mean. Such a model is countable from the outside but uncountable from the inside. — fishfry
On a different topic, let me ask you this question.
You flip countably many fair coins; or one fair coin countably many times. You note the results and let H stand for 1 and T for 0. To a constructivist, there is some mysterious law of nature that requires the resulting bitstring to be computable; the output of a TM. But that's absurd. What about all the bitstrings that aren't computable? In fact the measure, in the sense of measure theory, of the set of computable bitstrings is zero in the space of all possible bitstrings. How does a constructivist reject all of these possibilities? There is nothing to "guide" the coin flips to a computable pattern. In fact this reminds me a little of the idea of "free choice sequences," which is part of intuitionism. Brouwer's intuitionism as you know is a little woo-woo in places; and frankly I don't find modern constructivism much better insofar as it denies the possibility of random bitstrings. — fishfry
I have so many pennies in this hand, that many in that hand. How many do I have in all. If X is my left hand and Y is my right hand, then I have X+Y pennies. That's the truth of it, in the sense that X+Y = (X+Y), and the fact of it as expressed by the pennies themselves. So it appears that in appropriate use, algebra yields both truth and fact. Refute or yield. Keep in mind your statements are categorical and not conditional. Your refutation therefore should be on the same terms.If it is being used as a system of logic employed toward determining the truth, it is faulty because it has a false premise. — Metaphysician Undercover
Probably I don't understand the point of the conversation. But just to be clear. The space tessellation/partitioning was not to show how one mathematical construction can be derived from another. The tessellation corresponded to some unspecified physical roughness with uniformly spaced, but irregularly situated constituents. It aimed to illustrate that solving world-space problems imperfectly (due to efficiency constraints) results in the adoption of a modus operandi solution, whose own structure exists only in concept-space.Sorry simeonz, but your example seems to be lost on me. The question was whether whatever it is which is represented by √2 can be properly called "an object". You seem to have turned this around to show how there can be an object which represents √2, but that's not the question. The question is whether √2 represents an object. — Metaphysician Undercover
Again, I do not hold than there is such a thing as "an abstraction existing as an object." I reject your peculiar terminological stipulation that an "object" can only be something that ontologically exists.Where is your demonstration of an abstraction existing as an object, which is not a demonstration of Platonism? — Metaphysician Undercover
No, a symbol in logic is itself either a subject or the predicate within a proposition. If it is a subject, then it denotes an object, which can be an abstraction or an existent. If it is the predicate, then it signifies the interpretant, which is a relation among the objects denoted by the subjects.... I was talking about what is represented by the symbol in logic, and that is a subject, not an object. — Metaphysician Undercover
I have so many pennies in this hand, that many in that hand. How many do I have in all. If X is my left hand and Y is my right hand, then I have X+Y pennies. — tim wood
It aimed to illustrate that solving world-space problems imperfectly (due to efficiency constraints) results in the adoption of a modus operandi solution, whose own structure exists only in concept-space. — simeonz
In other words, we conceptualized indeterminacy, not because of its objective existence (aleatoric uncertainty), but due to our lack of specific knowledge in many circumstances and because introducing indeterminacy as a model was the most fitting solution to our problems. — simeonz
As I said, I may misunderstand the topic of the discussion altogether, which is fine. But just wanted to be sure that the intention of my example was clear. (That is - that the space tessellation was not intended as a mathematical structure, but as representation of some unknown coarseness of the physical structure, being ignored for efficiency reasons.) — simeonz
Again, I do not hold than there is such a thing as "an abstraction existing as an object." — aletheist
No, a symbol in logic is itself either a subject or the predicate within a proposition. If it is a subject, then it denotes an object, which can be an abstraction or an existent. If it is the predicate, then it signifies the interpretant, which is a relation among the objects denoted by the subjects. — aletheist
They do represent objects--abstractions, not existents.Then your beliefs are irrelevant to my concerns with algebra and set theory, which hold that the symbols represent objects. — Metaphysician Undercover
On the contrary, this is Semeiotic 101--in a proposition, the subjects denote objects, and the predicate signifies the interpretant.I'm afraid you have things backward. — Metaphysician Undercover
Incommensurability does not preclude (mathematical) existence. Our inability to measure two different objects (abstractions) relative to the same arbitrary unit with infinite precision does not entail that one of them is (logically) impossible.The problem I've been discussing is that whatever it is which is expressed as "a square" does not actually exist in "concept-space" because the perpendicular sides are incommensurable ... The figure is impossible, just like the irrational nature of pi tells us that a circle is impossible. — Metaphysician Undercover
Only according to your peculiar theory, not the well-known and well-established theory in question.The figures defined by the theory are impossible, according to the theory, just like a square circle is impossible. — Metaphysician Undercover
If you view mathematics as explanative device for natural phenomena, I can certainly understand your concern. However, I see mathematics first and foremost as an approximate number crunching and inference theory. I do not see it as a first-principles theory of the space-time continuum or the world in general. I see physics and natural sciences as taking on that burden and having to decide when and what part of mathematics to promote to that role. If necessary, physics can motivate new axiomatic systems. But whether Euclidean geometry remains in daily use will not depend on how accurately it integrates with a physical first-principles theory. Unless the accuracy of the improved model of space is necessary for our daily operations or has remarkable computational or measurement complexity tradeoff, it will impact only scientific computing and pedagogy. Which, as I said, isn't the primary function of mathematics in my opinion. Mathematics to me is the study of data processing applications, not the study of nature's internal dialogue. The latter is reserved for physics, through the use of appropriate parts of mathematics.The problem I've been discussing is that whatever it is which is expressed as "a square" does not actually exist in "concept-space" because the perpendicular sides are incommensurable. There is a deficiency in the concept which makes it impossible that there is a diagonal line between the two opposing corners, when there is supposed to be according to theory. The figure is impossible, just like the irrational nature of pi tells us that a circle is impossible. — Metaphysician Undercover
As I said above, I don't think that mathematics should engage directly to enhance our knowledge of the physical world, but rather to improve our efficiency in dealing with computational tasks. It certainly is a very important cornerstone of natural philosophy and natural sciences, but it is ruled by applications, not natural fundamentalism. At least in my view.The problem in this situation is that the indeterminacy is created by the deficient theory. It is not some sort of indeterminacy which is inherent in the natural world, it is an indeterminacy created by the theory. Because this indeterminacy exists within the theory, it may appear in application of theory, creating the illusion of indeterminacy in the natural thing which the theory is being applied to, in modeling that natural thing, when in reality the indeterminacy is artificial, created by the deficient theory.
We can allow the indeterminacy to remain, if this form of "concept-space" is the only possible form. But if our goal truly is knowledge, then it cannot be "the most fitting solution" to our problems. — Metaphysician Undercover
This is what I mean.I'm not sure I actually understood your example. Maybe we can say that Euclidian geometry came into existence because it worked for the practises employed at the time. People were creating right angles, surveying plots of land with parallel lines derived from the right angles, and laying foundations for buildings, etc.. The right angle was created from practise, it was practical, just like the circle. — Metaphysician Undercover
I do not see how our newfound knowledge about the universe will impact all of the old applications. How does it apply to the geometries employed in a toy factory, for example. The same computations can be applied in the same way. Unless there is benefit to switching to a new model, in which case both models will remain in active use. This is the same situation as using Newtonian physics instead of special or general relativity for daily applications. It is simpler, it works for relative velocities in most cases, and has been tested in many conventional applications. I am breaking my own rule and trespassing into natural sciences, but the point is that a computational construct can remain operational long after it has been proved fundamentally inaccurate. And therefore, its concepts remain viable object of mathematical study.Then theorists like Pythagoras demonstrated the problems of indeterminacy involved with that practise.
Since the figures maintained their practicality despite their theoretical instability, use of them continued. However, as the practise of applying the theory expanded, first toward the furthest reaches of the solar system, galaxy, and universe, and now toward the tiniest "grains" of space, the indeterminacy became a factor, and so methods for dealing with the indeterminacy also had to be expanded. — Metaphysician Undercover
Of course it would. I meant applications where the grain is indeed uniform, such as the atomic structure of certain materials. And even then, only certain materials would apply. The point being is - every construct which can be usefully applied as computational device in practice deserves to be studies by mathematics. As long as it offers the desired complexity-accuracy tradeoff.Now, to revisit your example, why do you assume "grain uniformity"? Spatial existence, as evident to us through our sense experience consists of objects of many different shapes and sizes. Wouldn't "grain uniformity" seriously limit the possibility for differing forms of objects, in a way inconsistent with what we observe? — Metaphysician Undercover
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