But you didn't say any of those things. Instead you just accepted the mathematical definition I repeatedly gave you, and then kept arguing from your own private definition. When I finally figured out what you were doing, I was literally shocked by your bad faith and disingenuousness. I'm willing to have you explain yourself, or put your deliberate confusion-inducing equivocation into context, but failing that I no longer believe you are arguing in good faith at all. You have no interest in communication, but rather prefer to waste people's time by deliberately inducing confusion. — fishfry
My God, you wield your ignorance like a cudgel. I could have just as easily notated the two ordered sets as:
* ({1,2,3,4,…},<)({1,2,3,4,…},<) and
* ({1,2,3,4,…},≺)({1,2,3,4,…},≺)
which shows that these two ordered sets consist of the exact same underlying set of elements but different linear orders. Remember that sets have no inherent order. So {1,2,3,4,...} has no inherent order. The order is given by << or ≺≺. — fishfry
By "shown" you do not mean "displayed". After all, "it is not visible". Therefore, you must not be making the argument - as you did earlier - that we see the order but do not apprehend it. — Luke
You are saying that the "exact spatial positioning" is logically demonstrated by
("shown" in) the diagram, but it is not apprehended? If the exact spatial positioning is not (or cannot be) apprehended, then how has it been logically demonstrated by the diagram? — Luke
Of course, location is intelligible, conceptual, so it's not actually sensed it's something determined by the mind, just like meaning. When you tell someone something, they do not sense the meaning. This is "showing" in the sense of a logical demonstration. And the point is that upon seeing, or hearing what is shown, the mind may or may not produce the required conceptualization, which would qualify for what we could call apprehending the principle. More specifically, the conceptualization produced in the mind being shown the demonstration is distinct and uniquely different from the conceptualization in the mind which is showing the demonstration, therefore they are not the same. That is why people misunderstand each other.. — Metaphysician Undercover
I answered the question, it's just like a logical demonstration. Like when one person shows another something, and the other makes a logical inference. All instances of being shown something intelligible, i.e. conceptual, bear this same principle. We perceive something with the senses and conclude something with the mind. The things that I sense with my senses are showing me something intelligible, so I produce what serves me as an adequate representation of that intelligible thing, with my mind. — Metaphysician Undercover
Inherent order is a wider concept, applying in particular to biological systems and natural phenomena. Ordering is more specific having to do with listing. I think you are discussing the latter. — jgill
There is a logical demonstration of an order. The order which the mind apprehends, based on the demonstration is not the same order as that which inheres within the things shown. You keep neglected the principal point of the argument, that the order apprehended in the mind is not the same as the order in the object. Therefore "the exact spatial positioning" is not what is being demonstrated. — Metaphysician Undercover
The inherent order is the exact spatial positioning shown in the diagram. — Metaphysician Undercover
What is "THE INHERENT" order you claim that the dots have?
— TonesInDeepFreeze
The one in the diagram. Take a look at it yourself, and see it. — Metaphysician Undercover
Here you say that "the exact spatial positioning" is not what is being demonstrated (i.e. shown) in the diagram. — Luke
Which is it? — Luke
Therefore "the exact spatial positioning" is not what is being demonstrated.
— Metaphysician Undercover
Here you say that "the exact spatial positioning" is not what is being demonstrated (i.e. shown) in the diagram.
— Luke
No that's not a good interpretation. — Metaphysician Undercover
You need to respect the fact that — Metaphysician Undercover
What is "THE INHERENT" order you claim that the dots have?
— TonesInDeepFreeze
The one in the diagram. Take a look at it yourself, and see it. — Metaphysician Undercover
I don't care about your latest position. In case you missed it, my entire point for the last three or four pages is that you changed your position three or four pages ago. This is obvious from the quote that you somehow managed to overlook: — Luke
When I said order is spatial and temporal, you claimed a completely "abstract order", which I didn't understand, and still don't understand because you haven't yet explained this in a coherent way. — Metaphysician Undercover
I haven't changed my position. You have not yet understood it. — Metaphysician Undercover
What is "THE INHERENT" order you claim that the dots have?
— TonesInDeepFreeze
The one in the diagram. Take a look at it yourself, and see it. — Metaphysician Undercover
order is inferred by the mind, it is not visible. — Metaphysician Undercover
The order is right there in plain view, as things are, but it is just not understood, because we do not have the capacity to understand it. — Metaphysician Undercover
The inherent order cannot be apprehended by us. — Metaphysician Undercover
the order is right there, in the object, as shown by the object, and seen by you, as you actually see the object, along with the order which inheres within the object, yet it's not apprehended by your mind. — Metaphysician Undercover
Therefore the order [is] in the mind it is not the order shown by the thing. — Metaphysician Undercover
order is inferred by the mind, it is not visible. — Metaphysician Undercover
The inherent order is the exact spatial positioning shown in the diagram. — Metaphysician Undercover
This is "showing" in the sense of a logical demonstration. — Metaphysician Undercover
"the exact spatial positioning" is not what is being demonstrated. — Metaphysician Undercover
An order that is shown can be seen: — Luke
But we cannot see the inherent order: — Luke
The idea, as with anything else mathematical, is that we have some aspect of the real world, in this case "order"; and we create a mathematical formalism that can be used to study it. And like many mathematical formalisms, it often seems funny or strange compared to our everyday understanding of the aspect of the world we're trying to formally model. — fishfry
After all bowling balls fall down, and the moon orbits the earth. To help us understand why, Newton said things like F=maF=ma, and F=m1m2r2F=m1m2r2. And E=12mv2E=12mv2, and things like that. And you could just as easily say, "Well this doesn't seem to be about bowling balls. These are highly artificial definitions that Newton just made up." And you'd essentially be right, while at the same time totally missing the point of how we use formalized mathematical models in order to clarify our understanding of various aspects of the real world. — fishfry
So if you can see the difference between a real world thing like order, on the one hand; and how mathematicians formalize it, on the other; and if you are interested in the latter, if for no other reason than to be better able to throw rocks at it, I'm at your service. — fishfry
That's the mindset for understanding how math works. You seem to object to math because it's a formalized model and not the thing itself, but that's how formal models and formal systems like chess work. They are not supposed to be reality and it's no knock agains them that they are not reality. They're formal systems. If you can see your way to taking math on its own terms, you'd be in a better position to understand it. And like I say, for no other reason than to have better arguments when you want to throw rocks at it. — fishfry
We are always seeing things without actually seeing them — Metaphysician Undercover
If the universe is endlessly expanding forever and ever isnt that an infinite scenario? Will it stop expanding? If not then the universe is infinite. If it does stop expanding could it have expanded forever if circumstances allowed it to. — Keith W
I follow this, it seems to be exactly what I've been trying to explain to Luke, so we're on the same page here. — Metaphysician Undercover
These are what I would call universals, generalities produced from inductive reasoning, sometimes people call them laws, because they are meant to have a very wide application. — Metaphysician Undercover
As inductive conclusions they are derived from empirical observations of the physical world — Metaphysician Undercover
The issue is with what you call the purely abstract. It appears to me, that you believe there are some sort of "abstractions" which are completely unrelated to the physical world. — Metaphysician Undercover
They are not generalizations, not produced from inductive reasoning, therefore not laws, or "artificial definitions", in the sense described above. — Metaphysician Undercover
You seem to think axioms of "pure mathematics" are like this, completely unrelated to, and not derived from, the physical world. — Metaphysician Undercover
I object to the parts of these formalizations which do not correspond with our observations of the world. — Metaphysician Undercover
These would be faulty inductive conclusions, falsities. — Metaphysician Undercover
You claim that they do not need to correspond, that they a completely unrelated to the physical world. — Metaphysician Undercover
Yet when you go to describe what they are, you describe them as inductive conclusions, above, which are meant to correspond, in order that they might accurately "clarify our understanding of various aspects of the real world.". — Metaphysician Undercover
So I see a disconnect here, an inconsistency. — Metaphysician Undercover
You describe "pure abstractions" as being related to the world in the sense of being tools, or formalizations intended to help us understand the world. — Metaphysician Undercover
Yet you insist that those who create these formalizations need not pay any attention to truth or falsity, how they correspond with the physical world, in the process of creating them. — Metaphysician Undercover
And you claim that when mathematicians dream up axioms, they do not pay any attention to how these axioms correspond with the world, because they are working within some sort of realm of pure abstraction. — Metaphysician Undercover
As an example consider what we've discussed in this thread concerning " a set". It appears to me, that mathematicians have dreamed up some sort of imaginary object, a set, which has no inherent order. — Metaphysician Undercover
This supposed object is inconsistent with inductive conclusions which show all existing objects as having an inherent order. — Metaphysician Undercover
You seem to think, that's fine so long as this formalized mathematical system helps us to understand the world. I would agree that falsities, such as the use of counterfactuals, may help us to understand the world in some instances. — Metaphysician Undercover
But if we do not keep a clear demarcation between premises which are factual, and premises which are counterfactual, then the use of such falsities will produce a blurred or vague boundary between understanding and misunderstanding, where we have no principles to distinguish one from the other. — Metaphysician Undercover
If axioms, as the premises for logical formalizations are allowed to be false, then how do we maintain sound conclusions? — Metaphysician Undercover
There's no contradiction here, I take it? — Luke
So how can it be seen? — Luke
From your side it must seem like you're being tag-teamed by Luke and myself, but I'm not reading his posts. I'm not aware of that half of the conversation. — fishfry
It's an abstraction intended to formalize an aspect of nature. If you think it's a generalization of something, you might be missing the point. Hard to say. — fishfry
I think you are missing the point. If I drop a hundred bowling balls and I say, "Bowling balls fall down. That's a law of nature," then THAT is an inductive conclusion.
But if you see 100 bowling balls fall down and you go, F = ma, that is an abstraction and a mathematical formalization. You don't seem to have a firm grasp on this. Do you follow my point here? — fishfry
Your notion of induction is wrong. "All bowling balls fall down," is an inductive conclusion. F = ma is a formalization. — fishfry
But there are non-physical parts of the world that we are interested in, such as quantity, order, shape, symmetry, and so forth. Those are the non-physical parts of the world that are formalized by math. — fishfry
Things in the world have order, and we have a mathematical theory of order that seeks to formalize the idea. — fishfry
On the other hand, of course there are non-physical, non-part-of-the-world abstractions too. Chess, for instance. Chess is a formal game, it's its own little world, it has a self-consistent set of rules that correspond to nothing at all in the real world. Knights don't "really" move that way. Right? Say you agree. How can anyone possibly disagree? — fishfry
But you are the one that insists that physical collections of things have an inherent order. And that's what the mathematical concept of order is intended to formalize. Things in the world have order, and we have a mathematical theory of order that seeks to formalize the idea.
Right? When mathematicians formalize numbers, they're abstracting and formalizing familiar counting and ordering. When they create abstract sets, they are formalizing the commonplace idea of collections. A bag of groceries becomes, in the formalization, a set of groceries. Surely you can see that. Why would you claim math is not based on everyday, common-sense notions of the world? — fishfry
Like what? Can you name some of these? Sets correspond to collections. — fishfry
But again I ask you, exactly WHICH mathematical ideas are not based on or inspired by the natural world? You must have something in mind, but I am not sure what. — fishfry
That's a useful mindset to have, so that we don't allow our everyday intuitions interfere with our understanding of the formalism. But of course historically, math is inspired by the real world. Even though the formalisms can indeed get way out there. — fishfry
The truth is in the thing. — fishfry
If I want to study the planets I put little circles on paper and draw arrows representing their motion. The truth is in the planets, not the circles and arrows. I hope you can see this and I don't know why you act like you can't. — fishfry
First, sets are intended to model our everyday notion of a collection. And in order to do a nice formalization, we like to separate ideas. So we have orderless sets, then we add in order, then we add in other stuff. If I want to put up a building, you can't complain that a brick doesn't include a staircase. First we use the bricks to build the house, then we put in the staircase. It's a process of layering. — fishfry
Our formalization begins with pure sets. It's just how this particular formalization works. — fishfry
If I represent a planet as a circle, you don't complain that my circle doesn't have rocks and and atmosphere and little green men. I'll add those in later. — fishfry
You act like all this is new to you. Why? — fishfry
I refer you to Galileo's sketch of Jupiter's moons. With this picture he started a scientific and philosophical revolution. Yet anyone can see that these little circles are not planets! There are no rocks, no craters, no gaseous Jovian atmosphere. Why do you pretend to be mystified by this obvious point? — fishfry
Do you feel the same way about maps? — fishfry
Tell me this, Meta. When you see a map, do you raise all these issues? — fishfry
I assume there are many different senses to the word "see". The word is used sometimes to refer strictly to what is sensed, and other times to what is apprehended by the mind. — Metaphysician Undercover
We sense things without apprehending what it is that is being sensed, as in my example of hearing a foreign language. — Metaphysician Undercover
Can you not hear foreign languages? This is a terrible analogy. This is something which we can perceive but cannot apprehend. Your analogy with molecules is equally bad, since it is something we can apprehend but cannot perceive. It is (or very recently was) your position that we can neither perceive nor apprehend the inherent order. Remember? You said that order is "not visible". — Luke
We sense things without apprehending what it is that is being sensed, as in my example of hearing a foreign language. — Metaphysician Undercover
we neither perceive nor apprehend the meaning in the foreign language — Metaphysician Undercover
Now you say that we neither sense nor perceive the meaning of a foreign language: — Luke
Do you have an attention deficit or memory disorder? You seem incapable of maintaining your own position. A moment ago you were talking about "the reality of the situation" that we see the inherent order without apprehending it, and now you've switched back to saying that we can neither see nor apprehend the inherent order. If you think there are two different senses of the word "see" involved here, then define them both. Because one of them seems to refer to what we cannot see, which is not a familiar definition of the word "see". — Luke
I don't see the problem. Do you not grasp a difference between hearing people talking, and apprehending the meaning? Meaning as analogous with order, was the example. — Metaphysician Undercover
A moment ago you were talking about "the reality of the situation" that we see the inherent order without apprehending it, and now you've switched back to saying that we can neither see nor apprehend the inherent order. If you think there are two different senses of the word "see" involved here, then define them both. Because one of them seems to refer to what we cannot see, which is not a familiar definition of the word "see". — Luke
You said that we sense a foreign language without apprehending it. — Luke
We sense things without apprehending what it is that is being sensed, as in my example of hearing a foreign language. — Metaphysician Undercover
I already explained in what sense we see the inherent order, and do not see it, just like when we look at an object and we see the molecules of that object. — Metaphysician Undercover
And if you cannot see what is right in front of you, you think it must be "hidden" from you, instead of considering the possibility that your eyes are actually sensing it, but your mind is just not apprehending it., — Metaphysician Undercover
order is inferred by the mind, it is not visible. — Metaphysician Undercover
Therefore the order [is] in the mind it is not the order shown by the thing. — Metaphysician Undercover
Have you rejected your claim that we can see the inherent order? — Luke
2. How do you reconcile this with your statements that order is not visible? — Luke
Don't worry about that, the conversations are completely different. Luke is on a completely different plane. — Metaphysician Undercover
I don't see the distinction you're trying to make here, between an inductive conclusion, and "an abstraction intended to formalize an aspect of nature". What do you mean by "formalize" other than to state an inductive conclusion. — Metaphysician Undercover
I see the majority of definitions as inductive conclusions. Either they are like the dictionary, giving us a formalization (inductive conclusion) of how the word is commonly used, or they are intended to say something inductive (state a formalization) about some aspect of nature. — Metaphysician Undercover
I think you are missing the point. If I drop a hundred bowling balls and I say, "Bowling balls fall down. That's a law of nature," then THAT is an inductive conclusion. — Metaphysician Undercover
I think it's you who is missing the point. I do not have a firm grasp on the distinction you are trying to make, because there are no principles, or evidence to back up your claim of a difference between these two. — Metaphysician Undercover
F=ma says something about a much broader array of things than just bowling balls. — Metaphysician Undercover
So one could not produce that generalization just from watching bowling balls, you'd have to have some information telling you that other things behave in a similar way to bowling balls. Mass is a property assigned to all things, and the statement "f=ma" indicates that a force is required to move mass. How can you not see this as an inductive conclusion? It's not just a principle dreamed up with no empirical evidence. In all cases where an object starts to move, a force is required to cause that motion. It might have been the case that "force" was a word created, thought up, or taken from some other context and handed that position, as being what is required to produce motion (acceleration), but this does not change the inductive nature of the statement. — Metaphysician Undercover
As I said, I really do not understand how a "formalization" as used here, is anything other than an inductive conclusion. So I do not understand how you think my notion of induction is wrong. Perhaps you should look into what inductive reasoning is, and explain to me how you think a "formalization" is something different. I think induction is usually defined as the reasoning process whereby general principles are derived from our experiences of circumstances which are particular. — Metaphysician Undercover
That such things are non-physical is what I dispute. How could there be a quantity which is not physical? "Quantity" implies an amount of something, and if that something were not physical it would be nothing. "Order" implies something which is ordered, and if there was no physical things which are ordered, there would be no order. And so on, for your other terms. It makes no sense to say that properties which only exist as properties of physical things are themselves non-physical. — Metaphysician Undercover
When you say "formalize" here, do you mean to express in a formal manner, to state in formal terms? — Metaphysician Undercover
If it is physical things in the world which have order, and mathematics seeks to express this order in a formal way, then how is this not making a generalization about the order which exists in the phyiscal world, i.e. making an inductive conclusion? — Metaphysician Undercover
How can I agree with this? Chess is a game of physical pieces, and a physical board, with rules as to how one may move those physical pieces, and the results of the movements. The physical board and pieces are not "nothing at all in the real world", they are all part of the world. — Metaphysician Undercover
What's with your motive here? Why do you insist on taking rules like those of mathematics, which clearly refer to parts of the real world, and remove them from that context, insisting that they do not refer to any part of the real world? Your analogy clearly does not work for you. The chess game is obviously a part of the world and so its rules refer to a part of the real world, just like quantity, order, shape, and symmetry are all parts of the real world, and so the rules (or formalities) of these also refer to parts of the real world. — Metaphysician Undercover
Yes, I agree with this here. Now the issue is how can you say that there is a collection of things which has no inherent order. — Metaphysician Undercover
If things in the world have order, and mathematicians seek to formalize that order, then where does the idea of "no inherent order" come from? — Metaphysician Undercover
That notion of "no inherent order" is obviously not derived from any instance of order, and if mathematicians are seeking to formalize the idea of order, the idea of "no order" has no place here. It is in no way a part of the order which things have, and therefore ought not enter into the formalized idea of "order". — Metaphysician Undercover
Have you lost track of our conversation? The idea of "no inherent order" is what we are talking about, and this is what I say does not correspond with our observations of the world. — Metaphysician Undercover
We observe order everywhere in the world. Sets do not correspond to collections, because any collection has an inherent order, existing as the group of particular things which it is, in that particular way, therefore having that order, yet as a "set" you claim to remove that order. — Metaphysician Undercover
I'll repeat. It's what we've been discussing, your idea of "a set", as a collection of things with no inherent order. Something having no inherent order is not based in, nor inspired by the real world, we don't see this anywhere in the world. We can also look at the idea of the infinite. It is not inspired by anything in the natural world. It is derived completely from the imagination. — Metaphysician Undercover
Let's try this. We'll say that a "formalism" relates to the real world in one way or another, and then we can avoid the issue of whether it is an inductive conclusion. We'll just say that it relates to the world. Now, can we make a category of ideas which do not relate to the real world? Then can we place things like "infinity", and "no order" into this category of ideas? But rules about quantifying things, and rules about chess games do relate to the real world, as formalisms. — Metaphysician Undercover
Can you see that these ideas are not formalisms, nor formalizations in any way? Because they are purely imaginary, and not grounded in any real aspects of the natural world, there is no real principles whereby we can say that they are true or false, correct or incorrect. — Metaphysician Undercover
They cannot be classed as formalizations because they do not formalize anything, they are just whimsical imaginary principles. — Metaphysician Undercover
To use your game analogy, they are rules for a game which does not exist. People can just make up rules, and claim these are the rules to X game, but there is no such thing as X game, just a hodgepodge of rules which some people might choose to follow sometimes, and not follow other times, because they are not ever really playing game X, just choosing from a vast array of rules which people have put out there. Therefore there is nothing formal, so we cannot call these ideas formalisms or formalizations. — Metaphysician Undercover
I disagree with your notion of truth. I think truth is correspondence, therefore not in the thing itself, but attributable to the accuracy of the representation of the thing. Identity is in the thing, as per the law of identity, but "true" and "false" refer to what we say about the thing. — Metaphysician Undercover
I think this is a completely unreasonable representation of "truth", one which in no way represents how the term is commonly used. We say that a proposition is true or false, and that is a judgement we pass on the interpreted meaning of the proposition. We never say that truth is within the thing we are talking about, we say that it is a property of the talk. or a relation between the talk and the thing. — Metaphysician Undercover
Take a look at your example. The bricks are never "orderless". They come from the factory on skids, very well ordered. Your idea of "orderless sets" in no way models our everyday notion of a collection. — Metaphysician Undercover
The point is that orderlessness is in no way a formalization. A formalization is fundamentally, and essentially, a structure of order. Therefore you cannot start with a formalization of "no order". This is self-contradictory. As I proposed above, the idea of orderlessness, just like the idea of infinite, must be removed from the category of formalizations because it can in no way be something formal. To make it something formal is to introduce contradiction into your formalism. — Metaphysician Undercover
What I'm complaining about is your attempt to represent nothing, and say that it is something. You have an idea, "no inherent order", which represents nothing real, It's not a planet, a star, or any part of the universe, it's fundamentally not real. Then you say that this nothing exists as something, a set. So this nothing idea "no inherent order" as a set. Now you have represented nothing (no inherent order), as if it is the property of something, a set. — Metaphysician Undercover
The idea of contradictory formalisms is not at all new to me. I am very well acquainted with an abundance of them. That's why I work hard to point them out, and argue against them. — Metaphysician Undercover
I don't see how this is analogous. Galileo represented something real, existing in the world, the motions of Jupiter's moons. What I object to is representing something which is not real, i.e. having no existence in the world, things like "no inherent order". This is not a representation, it is a fundamental assumption which does not represent anything. If a formalism is a representation, then the fundamental assumption, "no inherent order" cannot be a part of the formalism. — Metaphysician Undercover
Consider this analogy. The idea of "no inherent order" describes nothing real, anywhere. So why is it part of the map? Obviously it's a misleading part of the map because there is nowhere out there where there is no inherent order, therefore I would not want it as part of my map. — Metaphysician Undercover
Yes, I get very frustrated when the map shows something which is not there. I look for that thing as a marker or indicator of where I am, and when i can't find it I start to feel lost. Then I realize that it was really the maker of the map who was lost. — Metaphysician Undercover
Molecules are not visible to the naked eye. But we see the object, and the object is composed of molecules, therefore we must be seeing the molecules. — Metaphysician Undercover
You can be looking right at the stars, and see them all, therefore you are seeing the mentioned constellation, yet you still might not be able see that specific constellation. — Metaphysician Undercover
See the different senses of "see", and how "visible" might be determined based on the capacity of the observer, or the capacity of the thing to be observed? The inherent order is not visible to us, due to our deficient capacities, yet we do see it, because it exists as what we are seeing. — Metaphysician Undercover
1) We do not perceive order with the senses. No problem so far, as we understand order with the mind, not the senses. 2) We cannot apprehend the inherent order. Correct, because the order which we understand is created by human minds, as principles of mathematics and physics, and we assign this artificially created order to the object, as a representation of the order which inheres within the object, in an attempt to understand the inherent order. But that representation, the created order is inaccurate due to the deficiencies of the human mind. 3)The inherent order is the exact positioning of the parts, which is what we do not understand due to the deficiencies of the human mind. — Metaphysician Undercover
If I see 100 bowling balls fall down, "bowling balls always fall down" is an inductive conclusion. But F = ma and the law of Newtonian gravity are mathematical models from which you can derive the fact that bowling balls fall down. It's a physical law, meaning that if you assume it, you can explain (within the limits of observational technology) the thing you observe. — fishfry
But this is not an important point in the overall discussion. — fishfry
Ok. I don't think the definition of induction versus a formal model is super important here. But "bowling balls always fall down" is simply a generalization of an inductive observation, whereas the law of gravity lets you derive the fact that bowling balls fall down; and that in fact on the Moon, they'd weigh less. The latter is not evident from "bowling balls always fall down," but it is evident from the equation for gravitational force. — fishfry
This point is not central to the main point, which is that models must necessarily omit key aspects of the thing being modeled. — fishfry
Ok fine, then order is physical and the mathematical theory of order is an abstraction or model that necessarily misses many important real-world aspects of order yet still allows us to get some insight. That's the point of abstraction, which I already beat to death in my last post. — fishfry
I'm explaining to you that whatever your concept of physical order is, mathematical order is an abstraction of it, which is necessarily a lie by virtue of being an abstraction or model, yet has value just as a map is not the territory yet lets us figure out how to get from here to there. — fishfry
I've conceded your point, now that I understand what you mean by inherent order. — fishfry
It's an abstraction that necessarily includes SOME aspects of the thing being modeled and excluces OTHER aspects. Just as a street map includes the orientation of the roads but ignores the traffic lights. — fishfry
That's right. A map is correct about some aspects of the world and incorrect about others. It's an abstraction. — fishfry
Maps are imaginary principles and don't formalize anything? Do you see why I think you're trolling? — fishfry
If you would engage with my examples of maps and globes, I would find that helpful. — fishfry
And sets represent aspects of collections, which exist in the world. And they omit "inherent order," which for sake of argument I'll agree collections in the world have. — fishfry
Well that has nothing to do with anything. Maps don't show things that aren't there. The question is, how do you feel when a map omits things that ARE there, like wet lakes and rivers, cars, and the size and scale of the actual territory being modeled. — fishfry
Then you don't see it. — Luke
And you claimed earlier that we could not possibly see it, in principle — Luke
The law of gravity is the more general statement, saying all things with mass will fall down. The statement that bowling balls will fall down is more specific. Inductive reasoning is to produce a general statement from empirical observations of particular instances. So, the law of gravity as a general statement, is an inductive conclusion. And, bowling balls may or may not have been observed in producing that inductive law, but the law extends to cover things not observed, due to the nature of inductive reasoning, and the generality of what is produced. This is why inductive reasoning gives us predictive capacity. That mathematics is used to enhance the predictive capacity of inductive reasoning is not relevant to this point. — Metaphysician Undercover
It is important, because induction, by its nature, requires observation of particular instances. And you seem to be arguing that there is a type of abstraction, pure abstraction, which does not require any inductive principles. So it is important that you understand exactly what induction is, and how it brings principles derived from observations of particular instances, into abstract formulae. Do you see that the Pythagorean theorem for example, as something produced from practice, is derived from induction? — Metaphysician Undercover
It is the inductive conclusion, which allows for the derivation, the prediction, which you refer to. As it is a general statement, it can be applied to things not yet observed. It is not the mathematics which provides the capacity for prediction. mathematics enhances the capacity — Metaphysician Undercover
The central point is the difference between the inductive conclusion, which states something general, and the modeling of a "thing", which is a particular instance. At this point, we take the generalization, and apply it to the more specific. It must be determined how well the generalization is suited, or applicable to the situation. This requires a judgement of the thing, according to some criteria. — Metaphysician Undercover
I think your description of abstraction as missing things, is a bit off the mark. What abstraction must do is derive what is essential (what is true in all cases of the named type), dismissing what is accidental (what may or may not be true of the thing). Now, if order is essential to being a thing, then we cannot abstract the order out of the thing, to have a thing without order, because it would no longer be a thing. — Metaphysician Undercover
This is not true in a number of ways. First, good abstractions, inductive conclusions, or generalizations, do not lie because they stipulate what is essential to the named type. They speak the truth because every instance of that named type will have the determined property. — Metaphysician Undercover
Second, your proposed "mathematical order" is not an abstraction, inductive conclusion, or generalization. You started with the principle that there is a unity of things with no inherent order. — Metaphysician Undercover
So you have separated yourself from all abstraction, induction, or generalization, to produce a purely imaginary, and fictitious starting point. — Metaphysician Undercover
You cannot claim that the imaginary, and purely fictitious starting point, of "no inherent order" is a generalization, or an inductive conclusion, or in any way an abstraction of the physical order. You are removing what is essential to "order", by claiming "no order", therefore you have no justification in claiming that this is an abstraction of physical order. — Metaphysician Undercover
Do you recognize the difference between abstracted and imaginary? Imagination has no stipulation for laws of intelligibility, while abstraction does. — Metaphysician Undercover
OK, now lets proceed to look at your imaginary "mathematical order". — Metaphysician Undercover
Do you concede as well, that by removing the necessity of order from your "set", we can no longer look at the set as any type of real thing. — Metaphysician Undercover
Nor is it a generalization, an inductive conclusion, or an abstraction of physical order. It is purely a product of the imagination, "no order", and as such it has no relationship with any real physical order, no bearing, therefore no modeling purposefulness. It ought to be disposed, dismissed, so that we can start with a new premise, a proper inductive conclusion which describes the necessity of order. — Metaphysician Undercover
The idea of something with "no inherent order" is not an abstraction, as explained above. It is a product of fantasy, imaginary fiction. — Metaphysician Undercover
A map is not an abstraction, it is a representation. I see that we need to distinguish between abstraction, which involves the process of induction, producing generalizations, and as different, the art of applying these generalizations toward making representations, models, or maps. Do you see, and accept the difference between these two? We cannot conflate these because they are fundamentally different. The process of abstraction, induction, seeks what is similar in all sorts of different thing, for the sake of producing generalizations. The art of making models, or maps, involves naming the differences between particulars. These are very distinct activities, one looking at similarities, the other at differences, and for this reason abstraction cannot be described as map making. — Metaphysician Undercover
I was referring to the principles of "no inherent order", and "infinity", with the claim that these do not formalize anything. I wasn't talking about maps. — Metaphysician Undercover
The map analogy is not very useful, for the reason explained above, it doesn't properly account for the nature of inductive principles, abstraction. Generalizations may be employed in map making, but they are not necessarily created for the purpose of making maps. Now the map maker takes the generalizations for granted, and proceeds from there, but must choose one's principles. In making a map, what do you think is better, to start with a true inductive abstraction like "all things have order", or start with a fictitious imaginary principle like "there is something without order"? Wouldn't the latter be extremely counterproductive to the art of map making, because it assumes something which cannot be mapped? — Metaphysician Undercover
So, for the sake of argument, we can make the inductive conclusion, all collections which exist in the world have an inherent order. — Metaphysician Undercover
This is a valid abstraction, based in empirical observation, and it states that what is essential to, or what is a necessary property of, a collection, is that it has an inherent order. Do you agree then, that if we posit something without inherent order, this cannot be a collection? — Metaphysician Undercover
It doesn't have the essential property of a collection, i.e. order; therefore it is not a collection. — Metaphysician Undercover
Each map maker, based on the needs of that map maker's intentions, chooses what to include in the map. Abstraction, inductive reasoning, is very distinct from this, because we are forced by the necessities of the world to make generalizations which are consistent with everything. That's what makes them generalizations — Metaphysician Undercover
Perhaps, but I disagree. It's a matter of opinion I suppose. You desire to put a restriction on the use of "see", such that we cannot be sensing things which we do not apprehend with the mind. I seem to apprehend a wider usage of "see" than you do, allowing that we sense things which are not apprehended. So in my mind, when one scans the horizon with the eyes, one "sees" all sorts of things which are not "forgotten" when the person looks away, because the person never acknowledged them in the first place, so they didn't even register in the memory to be forgotten, yet the person did see them. — Metaphysician Undercover
No middle 'e' in judgment. I can't take anyone seriously who can't spell. — fishfry
The point is that the process of abstraction necessarily, by its very nature, must omit many important aspects of the thing it's intended to model. You prefer not to engage with this point. — fishfry
That's fantastic. The Pythagorean theorem is a beautiful, gorgeous, striking, brilliant, dazzling, elegant, sumptuous, and opulent example of an abstraction that is not based on ANYTHING in the real world and that has NO INDUCTIVE CORRELATE WHATSOEVER. I am thrilled that you brought up such an example that so thoroughly refutes your own point.
The Pythagorean theorem posits and contemplates a purely abstract, hypothetical, mathematical right triangle such that the sum of the squares on the legs is equal to the square on the hypotenuse. No such right triangle has ever, nor will ever, exist in the real world. — fishfry
Your own example falsifies this. I can never confirm the Pythagorean theorem in by observation of the world. I can only prove it deductively and never inductively. — fishfry
This is the complete opposite of induction. — fishfry
But getting back to the larger point: A map is a formalization of certain aspects of reality that necessarily falsifies many other things. Just as a set is a formalization of the idea of a collection, that necessarily leaves out many other things. I have a bag of groceries. I formalize it as a set. The set doesn't have order, the grocery bag does. The set doesn't have milk, eggs, bowling balls, and rutabagas, the grocery bag does. — fishfry
I think what you are saying is that an abstraction faithfully represents some aspects of the thing, and leaves out others. — fishfry
The essence of your argument is that "inherent order" is so tightly bound to the thing, that it can not be separated by an abstraction. I believe that's the core of our difference. Do I have that right? — fishfry
For purposes of the conversation, yes, I'll stipulate that. So what? It's how mathematical abstraction works. Just like the Pythagorean theorem does the same thing. — fishfry
Well now you're just arguing about my semantics. If I don't use the word generalization (which by the way I have not -- please note); nor have I used the phrase inductive conclusion. I have not used those phrases, you are putting words in my mouth. I said mathematical models are formalized abstractions, or formal abstractions. Or formal representations. That's what they are. The need not and generally don't conform to the particulars of the thing they are intended to represent. Just as an idealized right triangle satisfies the Pythagorean theorem, and no actual triangle ever has or ever will. Oh what a great example, I wish I had thought of it. Thank you! — fishfry
You say that like it's a bad thing! Ok, imaginary fiction then. But a useful one! — fishfry
So if you have a problem, it's your problem and not mine, and not math's. — fishfry
Well a set isn't actually a collection. A set is an undefined term whose meaning is derived from the way it behaves under a given axiom system. And there is more than one axiom system. So set is a very fuzzy term. Of course sets are inspired by collections, but sets are not collections. Only in high school math are sets collections. In actual math, sets are no longer collections, and it's not clear what sets are. Many mathematicians have expressed the idea that sets are not a coherent idea. It would be great to have a more sophisticated discussion of this point. I'm not even disagreeing with you about this. Sets are very murky. — fishfry
So therefore perhaps I should stop wasting my time trying to argue that math is inspired by reality, or that math is an abstraction of reality, and simply concede your point that it's utter fiction, and put the onus on you to deal with that. I could in fact argue the abstraction and inspired-by route, but you bog me down in semantics when I do that and it's tedious. — fishfry
Fuck you fishfry — Metaphysician Undercover
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