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

Let me remind you. You started in the discussion with the repeated assertion "sets have no inherent order". Check this post, I think you'll see that claim stated a number of times.

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

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. So I continued to insist order was spatial or temporal until you gave me examples of first and second place in competition, what I called order relative to best. I accepted this as non-spatial or temporal ordering, but I still don't see it as completely abstract because it still is based in concrete criteria for judgement.

You proceeded to define order in terms of "less than", as if you thought that this is purely abstract. However, I had already explained how "less that" is dependent on, defined in relation to, quantity. So you only contradicted your earlier claim that order is logically prior to quantity, by defining order in relation to quantity. And, since quantity is dependent on spatial separation between individuals you have not really escaped the spatial aspect of order, to get to a purely abstract order.

So this is where we stand. You have claimed a purely abstract order, but given me an order based in "less than" which is based in quantity. And quantity relies on spatial conception, so you have really given me a concept of order based in spatial conceptions..

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

Luke. I very consistently said, over and over again, that we do not see the order.

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

I went through that already, more than once. 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. So, do not ask again, this same strawman question. Check out these quotes:

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

Fishfry claimed that a set has no inherent order, and I questioned whether it is possible that there could be a thing with no inherent order.

i really do not think that "listing" is the subject here, because listing Is an ordering of symbols, not the things represented by the symbols. The list may represent an order, but the reason for the order is something other than the spatial order of the symbols. And fishfry insisted on the reality of a purely abstract order, which could not be a spatial relation of symbols, as listing is. We would need to find a principle of order which is purely abstract.

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

Here you say that "the exact spatial positioning" is not what is being demonstrated (i.e. shown) in the diagram. However:

The inherent order is the exact spatial positioning shown in the diagram. — Metaphysician Undercover

Here you said that the exact spatial positioning is what is being demonstrated (i.e. shown) in the diagram.

Which is it?

And as an indicator of how you continually change your position:

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

No that's not a good interpretation. You need to respect the fact that what is being shown to the observer, as inhering within the physical thing being used in the demonstration, is not the same order as that which exists in the mind of the person performing the demonstration. I said there is a demonstration of "an order". I also said "the order which the mind apprehends, based on the demonstration is not the same order as that which inheres within the things shown." Then I said "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." The exact spatial positioning is what inheres within the object, and though it is what is being shown by the one doing the demonstration, it is not the same order as what the person is trying to demonstrate. This is why I said fishfry's claim that the order was random is false. That's what the person doing the demonstration was trying to demonstrate, but it was not what the demonstration actually showed.

What is "being demonstrated" is an order which exists in the mind of the person making the demonstration. This is the first line, the "demonstration of an order". What appears to the person making the interpretation, as what is "shown", is the physical object with an inherent order. This is a representation of the order which exists in the mind of the person making the demonstration. It is not the same order, but a representation of it. So the order being demonstrated is not the same as the order which inheres within the representation, (as a representation is different from the thing it represents), and the order in the mind of the person interpreting what is shown, is not the same as the order which inheres in the object. And, because of this medium, which exists between the one demonstrating and the one interpreting, the physical object as symbols, the order on the minds of the two individuals is not the same. That as I said is why we misunderstand each other.

Which is it? — Luke

As I said, numerous times, the mind creates an order to account for the order assumed to be in the thing Therefore the order in the mind it is not the order shown by the thing. No change of position, just a difficult ontological principle to describe to someone with a different worldview.

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

What interpretation? It's exactly what you said.

You need to respect the fact that — 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:

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

This is so obviously the opposite of your updated position: that we cannot see the inherent order in the diagram. This is all the evidence I need to make my point. Of course, you won't admit to it because that's why you're here: to bullshit and troll people.

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

As I explained, I haven't changed my position. You have not yet understood it.

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

Ok. This is a good starting point.

The question is, are you interested in understanding mathematical order in a coherent way? 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.

So if you're interested, I can explain that. Or frankly the Wiki article can do the same. If you're interested. If not, not.

After all bowling balls fall down, and the moon orbits the earth. To help us understand why, Newton said things like $F = ma$, and $F = \frac{m_1 m_2} {r^2}$. And $E = \frac{1}{2} mv^2$, 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.

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.

And it's helpful to remember that the mathematical formalisms are not supposed to be reality. Nobody is saying they are. It's like chess. You don't complain about how the knight moves, because you understand that chess is a formal game that must be taken on its own terms.

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.

I haven't changed my position. You have not yet understood it. — Metaphysician Undercover

I have understood it. According to your latest position, the inherent order is the true order of actual physical objects which humans are unable to apprehend; akin to Kantian noumena. The order that we are able to apprehend - which is not the inherent order - is the apparent order, which is invented by humans and assigned to those objects "from an external perspective". The inherent order is not something that can be truly spoken, perceived or apprehended.

Although we can neither perceive nor apprehend the inherent order, you claim that it is not hidden.

But this only evolved into your current position at about this post, after you were pressed (and unable) to specify the inherent order. Your change in position is the reason for these contradictory statements:

We both can see and can apprehend the inherent order:

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

But we cannot see the inherent order:

order is inferred by the mind, it is not visible. — Metaphysician Undercover

And we can see the inherent order, but we cannot understand or apprehend it:

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

An order that is shown can be seen:

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

But an order that is shown cannot be seen:

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 exact spatial positioning is logically demonstrated in the diagram:

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

But the exact spatial positioning is not logically demonstrated in the diagram:

"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

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. The order is there, just like the molecules are there, and what our eyes are seeing, yet we do not distinguish nor apprehend the molecules nor the order, so we cannot say that we see it. We are always seeing things without actually seeing them, because it is a different sense of the word "see".

There was no change to my position, just a need to go deeper in explanation, to clarify the use of common terms, to assist you in understanding.

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

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.

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

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. As inductive conclusions they are derived from empirical observations of the physical world

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

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. They are not generalizations, not produced from inductive reasoning, therefore not laws, or "artificial definitions", in the sense described above. You seem to think axioms of "pure mathematics" are like this, completely unrelated to, and not derived from, the physical world.

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

I object to the parts of these formalizations which do not correspond with our observations of the world. These would be faulty inductive conclusions, falsities. You claim that they do not need to correspond, that they a completely unrelated to the physical world. 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.".

So I see a disconnect here, an inconsistency. 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. 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. 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.

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. This supposed object is inconsistent with inductive conclusions which show all existing objects as having an inherent order. 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. 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. If axioms, as the premises for logical formalizations are allowed to be false, then how do we maintain sound conclusions?

We are always seeing things without actually seeing them — Metaphysician Undercover

And this includes the inherent order? There's no contradiction here, I take it?

You have said that the inherent order can neither be perceived nor apprehended. So how can it be seen?

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

Language needs to reflect the scope of cosmological questions. It is reasonable that in the very long run, no matter how stable, all particles will decay. Then to make sense of those questions, wouldn't you want to redefine the universe as whatever energetic spacetime left after all ordinary particles have vanished into pure potential energy? Or are you only concerned with material matters and their relative forms?

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

Ok good. Progress is being made. One point, I am not reading the entire thread. 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.

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

Oh that's what you call universals. Physical laws? Ok. I'm not sure if that's standard but no matter. At least I have an idea now what you mean by that.

But "generalities produced from inductive reasoning?" I'm not sure if I agree with that. Surely F - ma is not a "generality" at all. On the contrary, Newton had to first define what he meant by force and mass. F = ma has sometimes been called a definition. 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.

As inductive conclusions they are derived from empirical observations of the physical world — 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.

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?

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

Well of course bowling balls are physical, and Newton was doing physics.

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.

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?

Perhaps that's why math is special. It's a formal game, but it's a formal game that seeks to model certain aspects of the world that are themselves not quite physical. Order, quantity, shape, symmetry.

They are not generalizations, not produced from inductive reasoning, therefore not laws, or "artificial definitions", in the sense described above. — Metaphysician Undercover

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?

You seem to think axioms of "pure mathematics" are like this, completely unrelated to, and not derived from, the physical world. — Metaphysician Undercover

How can you say that? Some of them obviously are. Most of them are. All of math is ultimately inspired by the world, just as the fictional story of Moby Dick was inspired by a real-world incident in which a ship was sunk by a whale. All fiction is inspired by the real world in one way or another, surely you know this.

I object to the parts of these formalizations which do not correspond with our observations of the world. — Metaphysician Undercover

Like what? Can you name some of these? Sets correspond to collections. Bags of groceries, baseball teams, solar systems. Cardinal numbers correspond to quantity, ordinal numbers to order. Group theory is the study of symmetry. Crystallographers study group theory.

What mathematical ideas don't have any correspondence or at least ultimate inspiration from some aspect of the real world?

These would be faulty inductive conclusions, falsities. — Metaphysician Undercover

Your notion of induction is wrong. "All bowling balls fall down," is an inductive conclusion. F = ma is a formalization.

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.

You claim that they do not need to correspond, that they a completely unrelated to the physical world. — Metaphysician Undercover

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.

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

Yes exactly. You say that like it's a bad thing! That's what formalization is. We have some aspect of the world, and we invent a mathematical formalization of it that captures its important features but that is distinct from the thing itself. So that we can use math and logic to draw mathematical conclusions, and use those conclusions to get insight about the original thing were were interested in.

So I see a disconnect here, an inconsistency. — Metaphysician Undercover

You aren't making your case. Surely you don't reject all of science because science builds mathematical models of certain aspects of reality, and that those models are not identical with the aspects of reality that they model.

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

Yes. As opposed to chess, say, which is a pure abstraction not intended to help us understand the world, but rather intended as an entertainment and pastime in and of itself.

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

Well as you know, math consists of logical implications. IF we assume this, THEN we may conclude that. We don't necessarily assert the truth of the antecedent. I think Bertrand Russell pointed this out. Because F = ma is not "literally" true of the world, it only formally represents certain aspects of the world. You have to be willing to make that conceptual split, between what is, on the one hand, and our abstract formalization, on the other. The formalization can never be true, because it's distinct from the thing it represents. The truth is in the thing. The formalization can't be true or false, it's not a thing in the world.

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

They pay a lot of attention to the suitability of the axioms for a given purpose. But in the end, the axioms must be lies, because they are not, and CAN NOT BE, identical with the things they represent.

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.

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

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.

This supposed object is inconsistent with inductive conclusions which show all existing objects as having an inherent order. — Metaphysician Undercover

The objects themselves that are in the world may well have inherent order. Our formalization begins with pure sets. It's just how this particular formalization works. I don't know why it troubles you. 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. Right? Do you reject representing planets as little circles on paper, devoid of features, even though the planets they represent have features? How on earth can we get science off the ground without the process of abstraction, in which we begin with only certain aspects of things, leaving other aspects out.

You act like all this is new to you. Why?

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

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?

If Galileo showed you this diagram would you complain that it's a lie because it doesn't show the features of Jupiter? It is "true" insofar as it faithfully represents the small aspect of reality that it's trying to model. The fact that Jupiter has moons was a huge, massive, world-changing discovery. That's what was important here. You reject this line of thinking entirely? When you go to the planetarium do you complain that those aren't the real planets, that the models are made of plastic and are too small and are therefore lies?

You can't be this obtuse. Are you trolling?

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

We are usually perfectly clear about these demarcations. When we look at a globe of the earth, we see the oceans but the oceans are not wet. We say to ourselves, "The real ocean is wet, but the ocean on the globe is made of hard plastic. It's a lie, but it's a lie in the service of the truth." Nobody in the world is confused by this but you!

If axioms, as the premises for logical formalizations are allowed to be false, then how do we maintain sound conclusions? — Metaphysician Undercover

Do you feel the same way about maps? Ever use a map? The map is not the territory. Yet the map shows us true things about certain aspects of the territory, like the street names and where the freeway is.

Tell me this, @Meta. When you see a map, do you raise all these issues? "The rivers aren't wet. The streets aren't filled with cars. It's made of paper." Well ok I can't remember the last time I saw a paper map. But you get the idea. A map is a representation of some aspects of the world that we find of interest. Maps are lies, of course. In fact maps ARE lies, since maps are flat and the earth is a sphere. The projection's all off. You know this, right?

Do you rail at maps, at planetariums. at Galileo's crude drawings that changed the world? Darwin draws a finch, and you say, "It doesn't cheep. It doesn't lay eggs. It doesn't eat worms. It's only a pencil sketch. It's a lie, it's a lie I tell you!" Do you do this? Frankly I doubt it. You only act this way to play a character on this site.

Bottom line: Abstraction is a process of capturing the essence of some aspects of a thing of interest, by leaving out all other aspects. Abstractions are necessarily lies because they must leave important things out. Yet from them, we discern truth.

There's no contradiction here, I take it? — Luke

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.

So how can it be seen? — Luke

I really don't know how, it's just the reality of the situation. We sense things without apprehending what it is that is being sensed, as in my example of hearing a foreign language. There is a matter of distinguishing the individual elements, one from another, which the sense organ does not necessarily do, despite sensing the elements together as a composite.

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

Don't worry about that, the conversations are completely different. Luke is on a completely different plane.

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 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.

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.

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

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.

F=ma says something about a much broader array of things than just bowling balls. 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.

Your notion of induction is wrong. "All bowling balls fall down," is an inductive conclusion. F = ma is a formalization. — fishfry

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.

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

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.

Things in the world have order, and we have a mathematical theory of order that seeks to formalize the idea. — fishfry

When you say "formalize" here, do you mean to express in a formal manner, to state in formal terms? 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?

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

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.

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.

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

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. 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? 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".

Like what? Can you name some of these? Sets correspond to collections. — fishfry

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. 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.

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

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.

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

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.

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. They cannot be classed as formalizations because they do not formalize anything, they are just whimsical imaginary principles. 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.

The truth is in the thing. — fishfry

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.

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

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.

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

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.

Our formalization begins with pure sets. It's just how this particular formalization works. — fishfry

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.

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

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.

You act like all this is new to you. Why? — fishfry

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.

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

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.

Do you feel the same way about maps? — fishfry

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.

Tell me this, Meta. When you see a map, do you raise all these issues? — fishfry

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.

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

I'm not familiar with any sense of the word "see" which means "not see".

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".

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

I don't think the analogies are bad. That there is order inherent within the thing seen is something inferred, just like that there is meaning in the foreign language which is heard, is something inferred, and that there are molecules in the object seen is something inferred. We neither perceive nor apprehend the inherent order but we infer that it is there, just like we infer that there is meaning in the foreign language, and that there are molecules within the thing seen. But we neither perceive nor apprehend the meaning in the foreign language, nor do we perceive or apprehend the actual molecules in the object seen. We apprehend a representation of the molecules, just like we apprehend a representation of the inherent order. And, when we come to understand the language we apprehend a representation of the meaning intended (what is meant) by the author.

Earlier, you said that we sense or perceive a foreign language without apprehending it:

We sense things without apprehending what it is that is being sensed, as in my example of hearing a foreign language. — Metaphysician Undercover

Now you say that we neither sense nor perceive the meaning of a foreign language:

we neither perceive nor apprehend the meaning in the foreign language — Metaphysician Undercover

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".

Now you say that we neither sense nor perceive the meaning of a foreign language: — 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.

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

You seem to be going through great effort to create problems where there are none. Oh well, it's what I've come to expect from you.

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

The problem is that you weren't talking about meaning before. You said that we sense a foreign language without apprehending it. Now you're talking about something else: that we can neither sense nor apprehend the meaning of a foreign language. What happened to your position from a day ago about being able to sense the inherent order?

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

According to your new position, we cannot see the meaning of a language. But you only introduced this analogy to support your claims regarding our ability to see the inherent order. I could now counter your new position, explaining how the analogy still does not work because we are able to apprehend a foreign language (by learning it), but we can never apprehend the inherent order. But why I should I bother? I'm tired of your constantly moving target. It's intellectually dishonest.

You said that we sense a foreign language without apprehending it. — Luke

To apprehend the language being spoken, is to understand the meaning. You work very hard to make understanding difficult for yourself.

We sense things without apprehending what it is that is being sensed, as in my example of hearing a foreign language. — Metaphysician Undercover

Keep spinning your bullshit.

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

This is what we were discussing before you changed the subject to the meaning of a foreign language.

You didn't get the molecule analogy so I went back to the language one, I believe I used it earlier. Now you claim not to get the language one, but that appeared to me like an intentional misinterpretation. You pretended as if you didn't understand that apprehending language is understanding meaning.

Each is an example of sensing something without apprehending what is being shown by the thing being sensed.

You think it must be "hidden", if we sense something without understanding it, but I think that idea is what's misleading you. It's not at all hidden, the mind is just lacking in the capacity to understand what is being sensed. Thinking it is "hidden" is a feature of your accusative nature. When you can't understand a person you blame the other, instead of introspecting your own capacity. 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.,

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

1. I note your recent change from talking about "seeing" to talking about "sensing". Have you rejected your claim that we can see the inherent order?

2. How do you reconcile this with your statements that order is not visible?

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

No, I think we see it in exactly the way that I explained.

2. How do you reconcile this with your statements that order is not visible? — Luke

I explained that. We see the object. The object exists as an instance of ordered parts, inherent order. Therefore we must be seeing the inherent order even though strictly speaking the order is not visible to the person who is seeing it. The "not visible" property is due to a deficiency in the capacity of the person who is seeing the order.

I used the molecule example. 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. That the molecules are not visible to the person seeing them is due to a deficiency in that person's capacities.

It's the same principle as when someone is pointing something out to you, and you're looking right at it, so you're definitely seeing it, because it's right there in your field of vision, yet you don't see the particular thing that the person is pointing out. Have you ever looked at stars, and had someone try to point out specific constellations to you? 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.

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. Go figure.

Don't worry about that, the conversations are completely different. Luke is on a completely different plane. — Metaphysician Undercover

Maybe I should start reading the rest of this thread so we can have a free-for-all instead of a tag team.

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

Good question, let me see if I can sharpen my explanation.

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.

But this is not an important point in the overall discussion.

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

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.

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

It's not an important point, but for what it's worth, I think you are missing a major point as to the nature of science.

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

It's not important to the larger conversation.

F=ma says something about a much broader array of things than just bowling balls. — Metaphysician Undercover

And yet it lets us derive the falling of bowling balls. After all this you DO understand the difference.

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

This point is not central to the main point, which is that models must necessarily omit key aspects of the thing being modeled.

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

Not central, let's move on.

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

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.

When you say "formalize" here, do you mean to express in a formal manner, to state in formal terms? — Metaphysician Undercover

One, to abstract, and two, to build a mathematical model of the abstraction. Or maybe those two are the aspects of the same process. I'm making a larger point, not splitting hairs.

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

Ok they are. What is your problem with this?

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

That's pure sophistry. There is no physical law that requires the piece to move the way they do.

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

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.

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

I don't say that. Now that I understand what you mean by order, I'm happy to agree that every collection of physical things has an inherent order, namely "where every item is in time and space."

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

Now that I understand what you mean by inherent order, I no longer need to argue this point. Physical collections have inherent order, if by order you mean "where everything is, or how everything is arranged, in time and space."

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

I've conceded your point, now that I understand what you mean by inherent order.

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

Once I concede that, what else have you got?

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

Now that I understand what you mean by order, I see what you are talking about and I no longer oppose your point. Simple matter of understanding what you mean by inherent order, which you could have, but inexplicably chose not to, explain many posts ago

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

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.

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

How do we make maps without drawing in the cars?

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

That's right. A map is correct about some aspects of the world and incorrect about others. It's an abstraction. It's a representation of SOME ASPECTS of the thing being modeled but by necessity not ALL aspects otherwise the map would have to be an exact copy of your entire city or state. Globes would have to be as big as the earth. That they are smaller means that they are incorrect regarding size. That the oceans on a globe are not wet means they are incorrect about the wetness of the oceans.

They cannot be classed as formalizations because they do not formalize anything, they are just whimsical imaginary principles. — Metaphysician Undercover

Maps are imaginary principles and don't formalize anything? Do you see why I think you're trolling?

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

You're just playing games now, not seriously engaging with me.

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

What would a true map be, in your opinion? A map of your city or town, say. Would it have to be the same size? Would the rivers and lakes have to be wet? Would the cars have to be on it?

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

Map map map map map. Engage with the point, please.

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

Map map map map map. Engage with the point, please.

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

Why are the oceans dry on globes? Engage with the point, please.

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

If you would engage with my examples of maps and globes, I would find that helpful.

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

If you would engage with my examples of maps and globes, I would find that helpful.

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

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.

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

I don't think I have anything left to say. Perhaps we're done. This isn't fun and it isn't educational.

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

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.

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

This is like saying that we can see infrared or ultraviolet light with the naked eye. We can't; not according to any common usage of the word "see".

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

Then you don't see it.

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

No, we simply don't see it. And you claimed earlier that we could not possibly see it, in principle:

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

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.

But this is not an important point in the overall discussion. — fishfry

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?

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

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

This point is not central to the main point, which is that models must necessarily omit key aspects of the thing being modeled. — fishfry

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.

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 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.

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

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.

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. So you have separated yourself from all abstraction, induction, or generalization, to produce a purely imaginary, and fictitious starting point. 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.

Do you recognize the difference between abstracted and imaginary? Imagination has no stipulation for laws of intelligibility, while abstraction does.

I've conceded your point, now that I understand what you mean by inherent order. — fishfry

OK, now lets proceed to look at your imaginary "mathematical order". 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. 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.

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

The idea of something with "no inherent order" is not an abstraction, as explained above. It is a product of fantasy, imaginary fiction.

That's right. A map is correct about some aspects of the world and incorrect about others. It's an abstraction. — fishfry

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.

Maps are imaginary principles and don't formalize anything? Do you see why I think you're trolling? — fishfry

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.

If you would engage with my examples of maps and globes, I would find that helpful. — fishfry

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?

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

So, for the sake of argument, we can make the inductive conclusion, all collections which exist in the world have an inherent order. 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? It doesn't have the essential property of a collection, i.e. order; therefore it is not a collection.

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

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

Then you don't see it. — Luke

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.

And you claimed earlier that we could not possibly see it, in principle — Luke

No, I think you misunderstood. Perhaps it was the use of "perceive" which is like "apprehend". I said we could not apprehend it with the mind, the mind being deficient. This does not mean that we cannot sense, or "see" it at all. But your limiting of "see", to only that which is apprehended by the mind, instead of allowing (what in my opinion is the reality of the situation) that we are sensing things which are not being apprehended by the mind, not "perceived", is making you think that just because we cannot apprehend it with the mind, therefore we are not sensing it at all.

I know it's a difficult issue and it appears as incoherency, as ontological issues often do, because they are difficult to understand, but I think we need to establish a separation between what is sensed, and the apprehension of it, to account for the differences between how different people apprehend very similar sensations.

I don't think the universe is infinite except in the sense that a circle has an infinite path. Events don't last so there is no infinity in the past. Every moment is really the next moment. The number of motions that may have happened is not fully infinite because motion of something material in motion is as well blurred within time.

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

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.

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

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.

@Meta this is such a great example. I wish I had thought of it myself. In soccer they call this an own goal, where you kick the ball into your own net and score a point against your own team.

Oh this is good. Just perfect. You made my day.

To make this clear: The exactitude of the Pythagorean theorem is FALSE for every actual right triangle that's ever existed. It's only in pure abstract mathematical space that it's true. So we go from a fact that is NEVER true in the real world, to one that is ALWAYS true in the abstract mathematical world. This is the complete opposite of induction. It's deduction. It perfectly shows the power of pure abstraction to reveal things about the real world while being based on nothing at all of the real world.

I drop a thousand bowling balls, they all fall down. "Bowling balls fall down." That's induction. I observe a thousand, a million, a gazillion, right triangles, and I note that the sums of the squares on the legs is NEVER equal to the square on the hypotenuse, but only sort of close. From that I DEDUCE -- not "induce," I DEDUCE -- that for a perfect, abstract, Platonic right triangle, the theorem is exact.

Meta you are secretly on my side. I knew it all along! Like a double agent I dispatched into the world long ago and forgot was secretly working on my behalf. I welcome you back to the world of pure, abstract mathematics, in which things can be deductively proven true that are NEVER inductively true in the real world.

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

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.

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

No middle 'e' in judgment. I can't take anyone seriously who can't spell.

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

I'm sorry, I can't focus. You so thoroughly demolished your own argument with the Pythagorean example that I can't focus on what you're saying.

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.

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

I think what you are saying is that an abstraction faithfully represents some aspects of the thing, and leaves out others.

So 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?

Let me say that again, because these posts are getting too long and I believe I've found the essence.

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?


In other words I could not separate, "Fat bearded guy in a red suit who flies around at Christmas time and climbs down chimneys," from the concept of Santa Claus, because the two notions are so tightly bound that to omit one is to forever de-faithfulize the representation.

Am I now understanding your point?

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

Not necessarily in the world, only in the formal model. Which is no problem.

So you have separated yourself from all abstraction, induction, or generalization, to produce a purely imaginary, and fictitious starting point. — Metaphysician Undercover

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. There is no right triangle in the world that obeys Pythagoras. Only fake, idealized, imaginary, formal, completely-made-up mathematical right triangles do. Euclid would have been glad to explain this point to you. There are no points, lines, and planes. They're pure mathematical abstractions inspired by, but very unlike, certain things in the real world.

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

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!

Do you recognize the difference between abstracted and imaginary? Imagination has no stipulation for laws of intelligibility, while abstraction does. — Metaphysician Undercover

Well, abstraction is inspired by things in the real world, and imaginary isn't. But both are instances of formal systems. For example a mathematical right triangle is an abstraction, and chess is imaginary.

OK, now lets proceed to look at your imaginary "mathematical order". — Metaphysician Undercover

As I just defined it, mathematical order is abstract and not imaginary, since it's inspired by the order found in nature.

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

My gosh, @Meta, have I ever in all the times we've been conversating EVER referred to sets as real things? They're abstract mathematical objects, hence "real" if by real you mean objects of human thought; as opposed to things in nature like rocks. Of course sets are not "real things." In fact unlike most mathematical objects , sets don't even have a definition. Nobody knows what a set is. A set is anything that satisfies the axioms of some set theory; and there are many distinct axiomatic set theories.

I would never call a set real. But I have never TRIED to call a set real. Why on earth do you think you're challenging me with such a silly question? "No longer" look at a set as real? I never did.

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

There are alternative foundations. I don't see how the choice of foundation is troubling you so much. If you don't like sets, try type theory. I'd say try category theory, but you can do set theory within category theory so that's no escape.

But of course that's not what you're saying. You are objecting to the mathematical concept of set. Well a lot of mathematicians have done the same. On far more sophisticated grounds, which is why it would help you to learn some math if you want to throw rocks at it.

But we conceive of sets as abstractions of collections; and for purposes of getting the formalization off the ground, we conceive of sets as having no order; and then we add the order back in via order theory. I truly don't see why you find this troubling, but I'll accept that you do.

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

You say that like it's a bad thing! Ok, imaginary fiction then. But a useful one! Functional analysis and differential geometry are based on set theory, and quantum physics and general relativity are based on FA and DG, respectively. So you can't deny the utility of set theory, even as you rail against its unreality. On the contrary, the unreality is the whole point of abstraction. But you deny it's abstraction. Ok then, fiction. Ok fine, here's SEP on mathematical fictionalism. There's a philosophical school of thought that completely accepts your premise that math is fiction, nevertheless an interesting and a handy one. That's pretty much the philosophy I'm expressing in my posts to you. Though to be fair, some days I'm a Platonist. Both points of view are useful.

So: Yes math is a fiction. A complete lie. Stuff someone dreamed up one day. What of it? It's still useful. Remember the great essay with the perfect title: "The Unreasonable Effectiveness of Mathematics in the Physical Sciences.' Doesn't that just say it all? Math is so fictional, so clearly NOT based on reality, that it's UNREASONABLE that it's so effective. Yet is is.

So nobody's disagreeing with your point. You need to get beyond your point that math is a fiction, to try to come to terms with why it's so useful.

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

You know, I see that I am no longer even trying to argue that math is based on reality or represents reality. I could, but then you'll just tangle me up in semantics and fine points. A stronger argument is for me to simply agree with you, completely and wholeheartedly, that math is fiction. And useful. So if you have a problem, it's your problem and not mine, and not math's.

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

Well one is hard-pressed to do physics these days without mathematical infinity, even though the world as far as we know is finite. And I take your point about order, that you think order is so tightly bound to "collections of things" that the two concepts can't be separated by any abstraction. But set theory falsifies that claim, since set theory DOES separate collection from ordered collection.

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

Well set theory isn't map making, of course. and so map makers should start by trying to capture the inherent order of the layout of the streets in a city. But set theorists don't have to do that. So the hell with the map analogy then.

Like I say you have now helped me to clarify my thinking. I have a much stronger position. Math is fiction, and it's useful, so what of it?

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

Yes. I'll stipulate that. And all right triangles in the world violate the Pythagorean theorem. Yet the mathematical version of collection, a set, need not and does not have inherent order; and mathematical versions of right triangles necessarily satisfy the Pythagorean theorem.

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 can't be a real-world collection, accepting your definition that the molecules in the ocean are "inherently ordered" by virtue of where each and ever one is at any particular moment. Likewise real world right triangles violate Pythagoras. Oh what a great example!

It doesn't have the essential property of a collection, i.e. order; therefore it is not a collection. — Metaphysician Undercover

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.

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

I never use the word generalizations. I say abstractions. But if you won't let me do that, then I'll retreat to, "Fiction, and so what?"

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

Was this for me? Oh I see that was for @Luke. LOL.

Well. I hope we can shorten this going forward. I think there are some key points.

* You think that inherent order is so tightly bound with the idea of collection, that the two notions can not be separated by any abstraction. Like Santa Claus and the fat bearded guy in the red suit. That's an interesting point.

* You think math is utter fiction. To which I say, Ok, I'm a mathematical fictionalist myself, and what of it? And Wigner makes the same point. Math is so clearly untrue, that it is unreasonable that it should be so effective. This should be a starting point for your thinking, not an end point. Yes math is fictional. I not only don't argue that point, I have been trying for years to get you to see that. You are the one who wants to reify it.

And that whether or not math is "really" a fiction, which frankly is doubtful, it is nonetheless highly useful to adopt that stance when trying to understand it, so as to take math on its own terms. If you try to figure out whether it's "real" you can drive yourself nuts, because the abstractions get piled on pretty high. So it's better just to take it as fiction and learn the rules. as you do when learning chess.

* 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.

I think these are the key points here.

No middle 'e' in judgment. I can't take anyone seriously who can't spell. — fishfry

What kind of petty bullshit is this? Fuck you fishfry, I thought we were trying to be civil with one another. I see you've gone off the deep end already, and it's only Monday.

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

I've engage with this point, explaining that I think it is wrong. If it's an important aspect, an essential feature, then if the abstraction processes "misses" it, the abstraction is wrong. If it is something which can be left out of the abstraction, it is in Aristotelian terms "accidental" or "an accident", and is not an important aspect. Abstraction separates the important from the unimportant, and if it omits important aspects it is faulty.

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

That's amazingly wrong, to think that the Pythagorean theorem is not based in anything from the real world. It's based in the method used to produced parallel lines for marking out plots of land. Check into the history of "the right angle", and you will learn this. Clearly this is something in the real world.

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

Huh? Construction workers prove the Pythagorean theorem in the real world, many times every day. Make a 3,4,5 triangle, tt never fails to produce the desired angle. How is this not proof? Try it yourself. Mark two points to produce a line. Use the Pythagorean theorem to make a right angle at each of the two points, and make two new points on those right angles, at equal distances from the original points. Measure the distance between the two new points, and you will see that it is the same as the distance between the two original points, and you have proven the Pythagorean theorem because you have used it to produce right angles, and have proven that the angles produced are in fact right angles by producing two more equivalent angles.

This is the complete opposite of induction. — fishfry

What you seem to not grasp, is that people were producing right angles long before the Pythagorean theorem was formalized. The Pythagorean theorem came into existence as a formalized description of what those people were doing. Therefore it is a generalization of what people were doing when they succeeded in producing the right angle, so it is an inductive conclusion. Try and see if you can apprehend pi as an inductive conclusion? It is a generalization, what all circles have in common, just like the Pythagorean theorem is a generalization, what all instances of "the right angle" have in common. If you produce an angle which is not consistent with what the Pythagorean theorem says, you have not produced the right angle.

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

As explained above, if an abstraction, or formalization, leaves out important aspects, then it is faulty. And if you insist on using the map analogy after I've explained why it is unacceptable, I will insist that if a map leaves out important things, then it is obviously a faulty map.

One reason why the map analogy is faulty, is because the map maker can decide, based on the purpose for which the map is being made, which aspects are important, and which are not. In the case of abstraction, formalizing, or generalizing, we have no choice but to adhere to the facts of reality, or else the formalizations will be incorrect.

I think what you are saying is that an abstraction faithfully represents some aspects of the thing, and leaves out others. — fishfry

An abstraction is a generalization. It does not represent "the thing" in any way, nor does it represent aspects of the thing. It represents a multitude of things, by creating a category or type, by which we can classify things. Again, another reason why the map analogy is misleading. It appears to make you think that an abstraction represents a thing, like a map does. That is incorrect, the abstraction is a generalization, a universal, which represents a multitude of things.

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

This is not really a good representation of my argument, because you don't seem to understand what abstraction is in anyway near to the way that I do. It's a good start anyway. But let me put it in another way. Let's suppose a category, or type called "thing". The abstraction, generalization, or formalization, would be a statement of definition, what it means to be a thing. This would be a statement as to what all things have in common, which makes it correct to call each of them a "thing". To be an acceptable definition, would be to be a good inductive conclusion. My argument is that the good inductive conclusion is that all things have inherent order therefore it would be a bad formalization, generalization, or abstraction, to posit a thing without inherent order because this is contrary to good inductive reasoning. Furthermore, I've argued that since such a principle, is not based in any inductive reasoning, it cannot truthfully be called an abstraction, generalization, or formalization, it is simply an imaginary fiction.

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

As I've explained, it is false to call this an abstraction. To make up a purely imaginary, fictitious principle, is not abstraction. And, the Pythagorean theorem is not at all like this. Creating the Pythagorean theorem was a matter of taking what people had been doing on the ground, producing the right angle and parallel lines, and using inductive reasoning to determine what all these cases of producing the right angle had in common. Therefore it is not a purely imaginary and fictitious principle, it is a truthful inductive statement about what all instances of the producing the right angle have in common.

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

Remember, you claimed a difference between a formalization, and an inductive conclusion. I did not accept such a difference, and asked you to validate this claim. You have not yet done so, but continue to speak as if your proposed distinction is a true distinction, while I have demonstrated that it is not. Therefore I suggest that you give up, as false, this claim to a difference.

You say that like it's a bad thing! Ok, imaginary fiction then. But a useful one! — fishfry

Yes, fictions are useful. The principal use of these is to mislead and deceive. A secondary use is entertainment, but this requires consent to the fact that what is presented is fiction.

So if you have a problem, it's your problem and not mine, and not math's. — fishfry

Of course, deception is a problem for the one being deceived, not the deceiver. Or maybe I'm just not entertained by your proposed entertainment. Again, still my problem, but perhaps you have made a poor presentation.

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

It was you who called a set a collection, and referred to some sort of mystical process of collecting, which allows for your proposed "no inherent order".

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

I dealt with this. Most math is not fiction, as evident in the example of the Pythagorean theorem. I differentiated the types of mathematical principles which are imaginary fictions though, things like "no inherent order", and "infinity".

Fuck you fishfry — Metaphysician Undercover

LOL. Should I read the rest? Second time today someone took one of my little jokes too seriously, I'll practice up on my smileys. :smile: :yikes: :cool: :rofl:

Say, did you know that the Pythagorean theorem is false in the real world? What do you make of that?

ps -- Ok I see you did respond to the rest. By claiming the Pythagorean theorem is literally true. I'll respond later to the entire post. But you know, there are no right angles in the real world. That you think there are is a problem. Must I really walk you through the basics of the philosophy of physical measurement? Sigh. And remember: Only one 'e' in judgment. I trust you won't make that error again. Smiley smiley smiley smiley smiley. Jeez.