This is simply not possible (where 'internal' applies to some self-organsing system). To recognise a system, a self organising one, there has to be an 'internal' and an 'external' otherwise you're just referring to 'everything', and a self organising system has to have a probability distribution function that is opposed to the Gaussian distribution, as this is just the definition of self-organising. — Isaac

I told you, systems theory is very flimsy, and I explained why. The exact separation, or boundary, between what is internal and what is external is impossible to establish, so that any proposed boundary is somewhat arbitrary, or sufficient for a specific purpose, but not a real boundary in any true or absolute sense.

So simply by the definition of a discrete system we've got, of necessity, an internal state, an external state, a Markov boundary, and two different probability functions on either side of that boundary. — Isaac

There is no such thing as a "discrete system" in nature. This is just a useful fiction. All natural things have other things overlapping them in space and time. The earth and sun overlap by gravity and radiation for example. My existence and my mother's existence overlap in space and time, as I come from her womb. And many other things overlap my existence.

The idea of discrete systems may be useful for some specific purposes, but not for describing living beings, or parts of living beings, because of the large degree of overlap. Von Bertalanffy described living beings as "open systems". An open system cannot be a discrete system because the environment is just as much a part of the definition ("open") as is the "system".

But I'm saying the regularities and rational relationships inhere within the conscious experience-of-the-world - so it's neither 'in the mind' nor 'in the world', and that this indicates a deep philosophical issue. — Wayfarer

This is what Plato pointed to years ago, and it is why there is no interaction problem for dualism. "Ideas" actually exist within the medium between soul and body, and are therefore themselves part of that very interaction. In Aristotelian hylomorphism, and consequently Marxist materialism, ideas consist of both a formal aspect, and a material aspect (content, subject matter).

Not how I use the word reason. — Isaac

OK, but "reason" was my word, and it was used synonymously with "why":

The physical instantiation is the model. the thing represented by that model is neurons. The point being that we cannot determine the reason (why) for the thing, through reference to the reason (why) for the representation. So we cannot determine whether the neurons act representatively, through reference to the model, because the model represents how the thing behaves, not the reason (why) for that behaviour. — Metaphysician Undercover

So you giving "reason" a different meaning was nothing but trying to argue through equivocation. And now you attempt to explain away that equivocation, by implying that you were trying to change the subject and create a distraction from what I was saying (to talk about something different, to use 'reason" in a different way).

The purpose of neurons is not to represent the outside world. — Isaac

You seem to have missed what I was saying to Banno, and just butted in to the conversation using words in a completely different way. In no way was I saying that the purpose of neurons is to represent the outside world. I was actually arguing the opposite to that, denying the reality of an outside world. I was saying that the human 'system' (if you'll allow me to use this word in an informal way), creates images, ideas, and such things within the mind, as symbols, which have meaning.

What I argued is that the symbols, (just like words for example), need not be in any way similar to any "external thing". Dreams are a good example. So the images within our minds whether they have sense input, or are derived through the dreaming mind, or some other creative way, relate to other things in a way which is analogous to the way that words, as symbols, relate to other things. There is no need for any sort of similarity between the symbol, and the thing "represented" by the symbol, constituting the meaning of the symbol.

Now, I do not think I even used the word "represent". The word I chose was "symbolize":

Any pattern could symbolize something. And not all symbols necessarily appear like symbols to everyone. — Metaphysician Undercover

But Banno asserted that it is known that neural nets are not representational. So my root word "symbol" got replaced by Banno's root word "represent", and you use Banno's word in a completely different way, to assume an "outside world".

To be clear, I stated earlier that there is no need to assume any "external world" at all, and extreme skepticism which doubts the reality of an "external" world. is well justified. To expand on this, I will say that it is highly possible that all of our relations with the so-called external world, including sensations, and communications with others, is done internally. It could well be the case that we simply model these activities as occurring through an external medium, but our models are backward, upside down, as often turns out to be the case (geocentrism for example), and all these activities actually occur through an internal medium. What is external to us may be absolutely nothing, a wall of nothingness could constitute the external boundary to one's body, as 'outer space', nothingness. Then the real medium, as that which lies between you and I as a separation, and through which we communicate, may be through the internal, 'inner space'.

If you don't accept the notion of abstract objects, then I admit that there's not much for us to discuss. — TonesInDeepFreeze

I cannot agree that abstractions are objects, unless we restrict "object" to refer only to abstractions. But then we could not use "object" to refer to anything else, or we'd have equivocation. And we would have to create a special form of the law of identity, such that when 'the same' abstraction exists in the minds of different people, we can still refer to it as "the same" abstraction, despite accidental differences between one person and another, due to different interpretations. The current law of identity requires that accidental differences would constitute distinct 'objects' which are therefore not the same, so we'd need a different law of identity.

If you do accept the notion of abstract objects... — TonesInDeepFreeze

If the law of identity describes what an "object" is, then I cannot accept the notion of abstract objects, for the reasons explained above. I might accept the notion of abstract objects, so long as you agree with me that the law of identity does not apply to this type of object, and we proceed with caution, so as not to equivocate between these two distinct types of objects, physical objects being described by the law of identity, and abstract objects being a different type of object to which the law of identity does not apply. Can you agree to that?

then I point out that a set theoretical intuition may begin with the notion of a thing being a member of another thing: The notion of membership. — TonesInDeepFreeze

Sure, "a thing" in this context, I assume is an abstract object, with no possibility for identity, so you can say whatever you want about "a thing", or "object". You can even say contradictory things about an object, because the object has no identity as a physical object does. You could say one thing about "the object" and I could say a contradictory thing about it, because there's nothing to ensure that we maintain consistency between the object in your mind, and the object in my mind, which both bear the same name. How would we even know that we're talking about the same object, except that we are using the same name? Do you propose that the name is the object? Then where is the abstraction?

So there are no restrictions in the sense of truth or falsity by correspondence, and to say contradictory things about the same object might be completely acceptable. The "thing" is a purely imaginary fiction, and we can use contradiction in fiction without a problem, though it might make the imaginary "thing" seem counterintuitive.

The notion of membership. — TonesInDeepFreeze

All right then, considering the above conditions, I'm ready to try and understand what "membership" means. What does it mean for one thing to be a member of another thing? Is it necessary that the thing which is a member be a different type of thing from the thing which it is a member of? If not, how would I distinguish between which things are the partakers, and which are partaken of?

That offers at least these prongs of refutation:

(1) I am mostly (but not exclusively) self-taught from textbooks; and textbooks in mathematics don't indoctrinate. Rather they put forth the way the mathematics works in a context such as presented in the book. A framework is presented and then developed. There is no exhortation for one to believe that the framework is the only one acceptable.

(2) Indeed, mathematics, especially mathematical logic, offers a vast array of alternative frameworks, not just the classical framework, including constructivism, intuitionism, finitism, paraconsistency, relevance logic, intensional logic .... And mathematics itself does not assert any particular philosophy about itself, as one is free to study mathematics with whatever philosophy or lack of philosophy one wants to bring to it.

(3) It is actually cranks who are narrowminded and dogmatic. The crank insists that only his philosophy and notions about mathematics are correct and that all the mathematicians meanwhile are incorrect. The crank doesn't even know anything about the mathematics yet the crank is full of sweeping denunciations of it. The crank makes wildly false claims about mathematics, and then doesn't understand that when he is corrected about those claims, the corrections are not an insistence that the crank agree with the mathematics but rather that the corrections merely point out and explain why what the cranks says about mathematics is untrue. It's as if the crank says, "classical music is all wrong because classical music never has regular meter" and then when it is pointed out that most of classical music does have regular meter, the crank takes that as narrow minded demand that he like classical music. And the crank is not even aware that mathematics, especially mathematical logic, offers a vast array of alternatives. Meanwhile, the crank's usual modus operandi is to either skip, misconstrue, or strawman the refutations and explanations given to him, thus an unending loop with the crank clinging to ignorance, confusion, and sophistry. — TonesInDeepFreeze

I disagree on a number of key premises here, so I do not consider any of this to be acceptable refutation. But that's all beside the point, just a difference of opinion on trivial matters.

Wrong again. — TonesInDeepFreeze

I'm glad to be wrong, of course. I actually enjoy discourse with you TIDF, you're generally well behaved and intelligent. I just don't see that you know any strong principles.

I don't need premises. I don't consider ants have bank accounts. I don't consider atoms have feelings. I can't for the life of me think why anyone would consider neurons having reasons for long enough to even consider the premises required. — Isaac

Well, if you consider that each and every internal organ has its own purpose, function, relative to the existence of the whole, which is a living being, then you would understand that each of these organs has a reason for its existence. If it has a function, it has a reason, that's plain and simple. It serves a purpose relative to the overall whole, which is the living being, therefore it has a reason for being there, to serve that purpose. Obviously, neurons serve a purpose relative to the existence of the being, therefore there is a reason for their existence.

I really do not understand how anyone could even consider denying this obvious fact. Those who do, seem to suffer from some form of denial which appears to be an illness.

Ha! But the notion that neurons have reasons is practically watertight? — Isaac

Yes, that neurons have reasons for being, based on their purpose, is a very sound principle. That neurons comprise a system is a very flimsy principle because they are actually a small part of a much larger "system", better known as a living being. And neurons serve a purpose (they have a function) relative to that being, but they do not make up an independent "system" in themselves.

If only we wrote posts on this forum as we would an article in a reputed philosophical journal! — Agent Smith

Some of us would not want to lower ourselves to that level.

They just fire according to physical laws, they don't have a reason. — Isaac

What about those "hidden states"? Those unknown aspects disqualify this conclusion. All you can say is that they act according to physical laws to an extent which does not include thos hidden aspects.

Why is there even a reason for the behaviour of neurons? They just fire according to physical laws, they don't have a reason. — Isaac

The point was, that you haven't the premises required to logically conclude that there is no reason for the behaviour of neurons.

We're back to the point where Banno started this, by claiming that the behaviour of neurons is not representative. I said you cannot conclude that without knowing the reason for the behaviour. To simply assert "they just fire according to physical laws" does not give that reason. You appear to assume that there is no reason. This is obviously an unsupported assumption, as the following example demonstrates.

All tools created by human beings operate according to physical laws, and this does not necessitate the conclusion that there is no reason for them. You are just proving my point, knowing how a thing operates (according to physical laws) does not provide you with the knowledge required to make any conclusion about why the thing operates that way. Denying that there is a reason why, is simply an uninformed, unjustified, and unwarranted assumption. So I've just gone around a circle, with you taking up where Banno left off, and proceeding back to Banno's starting point.

Put simply, a Markov boundary is the set of states which separate any system we're interested in studying from the parts we're not. — Isaac

Systems theory is extremely flimsy. Boundaries can be imposed for various reasons, and various degrees of arbitrariness, with varying degrees of openness and closedness. Then, on top of all these levels of arbitrariness, when things don't behave according to what is dictated by the imposed boundaries, we can just rationalize the misbehaviour through reference to mysterious things like "hidden states".

The mathematical approach is to assume that any object can be divided in any way, so there is an infinity of possible divisions for each thing to be divided. In physics though, the way an object can be divided is highly dependent on the composition of the object. — unenlightened

Yes, in physics the way an object "can be divided" is highly dependent on the composition of the object, but sometimes this fact is ignored by physicists. That's the problem I referred to with the way that frequencies and wavelengths are treated. The problem appears to be that there is no identified medium within which the electromagnetic waves exist, so the real properties of the waves cannot be determined or described, and physicists are left with theoretical mathematics which assumes an infinity of possible divisions.

But mathematicians have no mercy, and maths is full of irrationality ever since Pythagoras. Irrational numbers are the devil in the detail that he proved the existence of geometrically, and the fact that mathematicians (and others) are still trying to insist that maths should be fitted within the limits of their thinking is more to do with psychology than mathematics. — unenlightened

I would say that if the geometry can prove the existence of irrational numbers, then this is an indication that the geometry is faulty. The issue with pi and the square root of two appears to involve the traditional geometer's use of distinct dimensions. So the irrational nature of the square root of two shows that any two distinct dimensions, produced by the right angle, are fundamentally incompatible. Set two new points, equidistant from a starting point, on two traditional dimensions (making a right angle), and the distance between those two points will always be indefinite (an irrational number).

The issue with pi is very similar. A straight line has one dimension, and a curved line has two dimensions. These two types of lines are fundamentally incompatible. Some might say that an infinitely large circle, has an arc so gradual that it's actually a straight line, but how would you ever get two dimensions, a curve, or a circle then? Likewise, some might say that a polygon with infinite sides is actually a circle, but this doesn't really reconcile the incompatibility, because each side is still a one dimensional line, at an angle to the next side, it's sides are distinct straight lines and that's not a curved line.

I like your example of the taxicab geometry. I'm not sure, but it appears to deny the reality of the one dimensional straight line. But it appears to me, like if we deny the straight line, then we'll need an infinite number of dimensions, because there would be an infinite number of possible ways to get from one point to another point. So this doesn't really get rid of the irrationality.

This is talking about how neural network models might represent neurons, not how the physical instantiation of those models represent the external world. — Isaac

The physical instantiation is the model. the thing represented by that model is neurons. The point being that we cannot determine the reason (why) for the thing, through reference to the reason (why) for the representation. So we cannot determine whether the neurons act representatively, through reference to the model, because the model represents how the thing behaves, not the reason (why) for that behaviour.

Th neural net is not making a model that you then see with your mind. It is your mind seeing. — Banno

No the neural net is not the same as your mind seeing, because "seeing" obviously requires the activity of the eyes as well, the rods and cones for example. Otherwise seeing and dreaming might be the same thing. But REM is a different type of eye involvement. And we do know what other features are required as well, (the role of memory in seeing?). So the activity of the neural net cannot be said to be "your mind seeing". The "neural net" is a model which people have constructed and some might claim it to represent "your mind seeing", but it is a deficient model.

Spot on. — Isaac

Spot off, as explained above.

No, they are not built to represent a thing. It's simply not what they do. — Isaac

Banno loves Wikipedia, so:

The central connectionist principle is that mental phenomena can be described by interconnected networks of simple and often uniform units. The form of the connections and the units can vary from model to model. For example, units in the network could represent neurons and the connections could represent synapses, as in the human brain. — Wikipedia, Connectionism

That's explicit. representative models. I wouldn't exclude the possibility that Wikipedia is wrong though.

Sorry Banno, but you're not making any sense at all. There is a vast multitude of different ways of representing. That a process is not representational in a specific way, does not mean that it is not representational in an absolute way.

Connectionist models are just representations, so they are built for a reason other than the reason for which the thing represented was built, as they are built to represent that thing.

And I'll tell you one more time, knowing how the thing works allows you to build a representative model of it. But this provides you with no information as to whether or not the thing represented is itself a representation. A child can copy a word without even knowing that it's a word. To determine whether the action is representational requires a knowledge of why the thing is doing what it does. And that is completely different from why the model of the thing does what it does, because they are produced for different reasons.

Knowledge of how neural nets work will not inform you as to whether or not they are representational. This would require knowledge of "why". Knowing how the human vocal chords produce the sounds which are words, for example, does not provide you with information as to whether or not the sounds are representational. Again, this is fundamental to "meaning".

So the supposed 'external cause' of the sensation is not a cause at all, and cannot even be truthfully said to be a necessary condition. Therefore it is not at all irrational to be skeptical of the reality of a proposed 'external world'.

This isn't addressing the issue. Your eyes and my eyes are stimulated by the same light, reflected by the same external world source. Yet I see red and you see green. If "red" and "green" refer to some hidden state in the external world cause then what does it mean for me to "see red" and you to "see green" in this situation? The "red" and "green" are referring to some quality of our experiences. — Michael

The sensation of colour is best described by reference to the activities of the "cones" and the "rods" of the eyes, rather than reference to some external source. But even this does not account for the huge role of brain activity. Since we dream in colour, it may actually be brain activity which causes the sensation of colour. The role of REM in relation to dream imaging is not well understood.

Weightings in neural networks. You are thinking in terms of brains containing representations, but neural nets are not representational. — Banno

Why do you say that? Any pattern could symbolize something. And not all symbols necessarily appear like symbols to everyone. So I don't see any basis for the claim that "neural nets are not representational". To me, it looks far more likely that they actually are, as any ordered structure can be said to be representational. That's fundamentally what "ordered" is, a representation of the intent which orders, in other words, meaning.

It doesn't mean the rock doesn't fall through, or that anything even particularly different happens to it there. — noAxioms

It means that the concept of a rock falling through the event horizon, is incoherent. In other words, the theories applied, mathematics applied, or both, are faulty, because they produce an incoherent scenario.

Sure, you and I and the bird act towards some of the external world-mess of wave-particles as if they are eggs; but that does not mean that there are no eggs. Exactly the opposite. Other examples may make the point clear: money and mortgages and property and universities only exist because we act as if they exist; and yet it would be wrong to suppose that therefore they are just imaginings. "just" does not do them justice. — Banno

The problem though, is that the thing created by the mind, or brain, (I'll call it an image for now), need not even be at all similar to the supposed aspect of the real world which we assume that it represents. And minds or brains also created the "mess of wave-particles" as a type of image. We ought not assume that anything created by the mind or brain is in anyway similar to the external, real world.

For example, we know that the word "egg", and the word "cup", have no real similarity to the supposed real world things that the words represent. So why would we think that the visual image that the mind or brain creates, which we assume as a representation of a real world thing, is in any way similar to what the real world thing actually is. Just like a word, the visual image might be simply a convenient symbol.

don't think that follows at all. Electromagnetic radiation with a wavelength of ~650nm consistently triggers the experience of the colour red. — Michael

Things are not so simple. We see the object as red, we do not see the radiation as red. The object interacts with radiation which is everywhere around it, and emits 650nm, and this allows us to see it as red. I believe this is important to acknowledge, it is not really the radiation which triggers the experience, just like it is not the hammer which drives the nail. The radiation is a medium, and whatever it is, which is going on within the object, that is what is seen, just like the hammer is a medium, and the person swinging it is actually driving the nail.

Stand near a warm stove for example, and feel the radiant heat. What you are feeling, is the stove directly interacting with your body, despite the fact that this interaction is modeled by science as occurring through the means of some mysterious substance called "radiation". The problem is that we model the two distinct objects as distant from each other, separated by "space", and it is very counter-intuitive to think that these two objects could be directly interacting with each other, because there appears to be "space" between them. But in reality, we know that objects which appear to be separated by "space" actually do interact with each other directly, through gravity.

So the objects actually overlap each other, in space, and occupy the very same space as each other. Therefore we ought to recognize that this whole way of modeling objects as separated from each other by "space", and assuming a mysterious medium between them, "radiation", and saying that the radiation is what we sense, rather than sensing the object itself, directly, is fundamentally faulty.

With Trump, you know he left a calling card.

Well, that went over my head, but what caught my eye is that what you seem to be saying is our options aren't ∞∞ i.e. we may restricted to finite mathematics, even if not all the time, some of the time. — Agent Smith

What I think, is that we allow "infinite" so that we will always be able to measure anything. If our numbers were limited to the biggest thing we've come across as of yet, or largest quantity we've come across, then if we came a cross a bigger one we would not be able to measure it. So we always allow that our numbers can go higher, to ensure that we will always be able to measure anything that we ever come across. In that way, "infinite" is a very practical principle.

Don't go messing' with mathematicians with your "c'mon, be reasonable" attitude; they'll have none of it. — unenlightened

I guess that's why TIDF split the scene. But we were just playing, it's nothing harmful.

Initially I was of the opinion that infinity had to be replaced by an extreme number like Graham's number but the result ∑∞n=1=−112∑n=1∞=−112 (used in string theory) is evidence the number that we swap infinity with in a calculation, surprise, surprise, doesn't have to be large, a wannabe infinity; a small, nevertheless most special number like −112−112 will do just fine. — Agent Smith

Fractions are actually very tricky. The common approach is to assume that any object can be divided in any way, so there is an infinity of possible divisions for each thing to be divided. In reality though, the way an object can be divided is highly dependent on the composition of the object. So producing ratios in theory, and applying them without proper standards as to the true way that things can be divided, can be very misleading.

This becomes quite evident in wave theory, as the ancient Pythagoreans who studied harmonies and the properties of musical tones found out. There is a very real problem with the assumption that time (hence frequency) can be divided arbitrarily (in any way that one wants). This produces the uncertainty principle of the Fourier transform.

An object that has no members is either the empty set or an urelement. And of course, an object that has in it only one object is a non-empty set. — TonesInDeepFreeze

An "object" is one. To allow that an object has members requires a special definition. We can say that this object is a set, and define a "set" as a collection of objects. But to say that there is a collection of objects with no objects is contradictory, whether you admit to this fact or not.

It seems that you do not recognize the fundamental distinction between one and many, and you are now trying to reduce many to one, by saying that a set is an object. One is not a plurality, and a plurality is not one. That is a basic self-evident truth. If you want to talk about a category, or class of objects, this is something completely different. We cannot name the category "an object", and also name the members "objects", without equivocation. Do you apprehend the category mistake, and consequent fallacy of equivocation which occurs if we call both the category, and the members within that category, by the name "object"?

Talking to you is like conversing with a computer. — Agent Smith

I thought for a while, that Tones was a bot, endlessly repeating the same thing over and over without grasping the logical problems with what was repeated. Then I realized that this is simply the way mathematics is. The students are taught very specific principles, and their minds are funneled down a very narrow path, You could say that they are programmed, like a computer is programmed, and the possibility that the program is not a very good program, so that they are being misled, is excluded from the program. The students are discouraged from looking outside the program, and seeking the truth, like philosophers do, because truth is not important to mathematics.

You are entirely ignorant of what contradiction is in mathematics. — TonesInDeepFreeze

OK, Mathematics is allowed its own special definition of "contradiction", so that statements which would qualify as contradictory in a rational field of discipline do not qualify as contradiction in mathematics.

Second, even informally, you mention a certain definition of 'set'. Mathematicians are not then obliged to refrain from having an understanding in which "collection of objects" does not preclude that it is an empty collection of objects, notwithstanding that that seems odd to people who have not studied mathematics, and so more explicitly we say, "a set is a collection, possibly empty, of objects". You are merely arrogating by fiat that your own notion and definition must the only one used by anyone else lest people with other notions and definitions are wrong. That is an intellectual error: not recognizing that definitions are provisional upon agreement of the discussants and that one is allowed to use different definitions in different contexts among different discussants. It's like someone saying "a baseball is only one such that is used in major league baseball" and not granting that someone in a different context may say, "By 'baseball' I include also balls such as used in softball". It is intellectually obnoxious not to allow that. And it is one in the deck of calling cards of cranks. — TonesInDeepFreeze

When you explain to me how a set which has no objects also has a collection of objects, and this is not contradictory, then I'll start to listen to you.

So please explain to me, your understanding of "collection of objects" in which there is no objects. As far as I can tell, either you have a collection of objects, or you have no objects, but to have both is clearly contradictory. What if you had one object? It is neither a collection of objects nor is it no objects. Do you agree? Or do you just abandon rationality for the sake of mathematics?

I've studied the law of identity for quite some time. It is rather perplexing, and requires a good deal of thought to properly understand. It is stated as 'a thing is the same as itself'. Despite the appearance of meaninglessness, there are a number of facets to this law, which we can consider.

The first thing is that this law puts the identity of the thing within the thing itself. This is important to understand, because it means that a thing's true identity is not what we, as human beings say it is, it's true identity is what it is, itself. So if we say "that person is a man", "man" is not that thing's true identity. And even if we say "that person is Art48", "Art48" is not that thing's true identity, because the thing's true identity is the thing itself. The thing itself is the thing's identity

The next thing is that the law of identity allows that a thing might be continuously changing, yet maintain its status as the same thing. This is very difficult to conceive of, because normally when a thing changes, we would say that it ceases to be what it was, and it becomes something different. But since we've allowed that the thing's true identity is within the the thing itself, then what we say about the thing is irrelevant to the thing's identity. So despite the fact that we might say that the thing has changed from being this, to being that, within itself the thing has maintained its identity, as itself, and it may continue to be the same thing which it was.

The law of identity may be denied, as I believe Hegel did. But doing this renders the other laws of logic, noncontradiction, and excluded middle, as useless. This is because these laws put restrictions on what we can truthfully say about a thing, by determining what is impossible for us to truthfully say about a thing. However, if a thing has no identity within itself, then there is no such thing as a thing, and it would be meaningless to talk about what we can truthfully say about a thing.

There is no magic. Very much to the contrary. At a bare minimum, it is algorithmically verifiable whether a given formal expression is well formed and then whether a given sequence of formulas is a formal proof. That is a courtesy given by formal logic that is not hinted at in various handwavings and posturings by cranks as often found in a forum such as this. And I have given extensive explanation of many of the formulations I have mentioned. — TonesInDeepFreeze

Whether an expression is well formed or not is irrelevant to whether it is self-contradictory, because to determine contradiction we must analyze the meaning, and this is the content, not the form.

Yet no one who says things like the above has ever demonstrated that Zermelo set theoretic infinitistic mathematics implies a contradiction. — TonesInDeepFreeze

Have you forgotten the conversations we've had earlier? The empty set for instance, involves contradiction.

Yet, again, we remind that a contradiction is a statement of the form "P and not-P" — TonesInDeepFreeze

Oh, now I remember, you have a very odd notion of what constitutes contradiction, and this is how you insist that there is no contradiction even after contradiction is demonstrated to you. If the statement doesn't explicitly say "P and not-P", then there is no contradiction in your interpretation, regardless of what the statement means.

The law of noncontradiction states that the same object cannot both have and not have, the same property, at the same time, in the same respect. So consider the empty set for example. For simplicity, let's say that a set is a collection of objects. Therefore a set necessarily has objects. The empty set has no objects. Therefore "empty set is self-contradicting. The empty set is said to have objects (necessary to being a "set", or collection of objects) and also to not have objects (necessary to being empty), at the same time, and in the same respect

Agent Smith is poking at us. He is much more intelligent than he seems. — jgill

Gives TonesInDeepFreeze 8 mg of Zofran. — Agent Smith

TIDF can be very entertaining that way, and also quite enlightening. But eventually it gets boring when TDIF refuses to divulge the secrets of the mathemagician's smoke and mirrors.

In a sense then ∞∞ in science has a job description similar to contradictions - separates the possible from the impossible. — Agent Smith

This is actually a very good point. Every time infinity is employed in the application of mathematics, it's like employing a contradiction. This becomes very clear in an analysis of the common mathematician's claim to have resolved Zeno's paradoxes.

You're just restating the same point, that one ought not doubt everything, such a doubt is irrational, because "to doubt there must be one who doubts". But as I said, showing that such doubt is irrational does not demonstrate that it is logically impossible. Therefore it is you who is actually missing the point by refusing to accept that this sort of irrational uncertainty is a very real part of human life.

And, as I explained with those 750 wasted words, the point you are missing is very important epistemologically, as Socrates demonstrated. Because you miss this point, you will continue with your belief in platonic realism, falsely assuming that some non-temporal, intelligible objects of absolute certainty, underlie all our knowledge as a foundation to it. And, you'll be attracted to faulty theories, like the theory of recollection expressed in Plato's Meno, to support your false belief.

Actually, it's what you posted which is beside the point. Look, Augustine demonstrates that one "ought not" doubt such things, just like Wittgenstein claimed to have demonstrated such doubt to be irrational.

So whoever acquires a doubt from any source ought not to doubt any of these things whose non-existence would mean that he could not entertain doubt about anything. — Augustine, On the Trinity 10.10.14 quoted in Richard Sorabji Self, 2006, p.219

The point though, is that claims that we cannot have this sort of doubt, it is logically impossible, because doubting, it is claimed, presupposes certainty. And that is what I objected to, the claim that doubting presupposes certainty, so that such doubt is logically impossible. That someone might demonstrate such doubt as irrational, or another might claim that one ought not doubt some specific things, is beside the point. The point being that such doubt is a very true aspect of reality, whether it's irrational or not, as many living beings, humanity included, commonly do irrational things.

If he doubts, he knows that he does not know. — Augustine, On the Trinity 10.10.14 quoted in Richard Sorabji Self, 2006, p.219

It is a mistake to classify doubt as a type of knowing. Doubt is a form of uncertainty, and knowing is a form of certainty. So in relation to any particular subject, one effective excludes the other. Uncertainty, doubt, is a lacking, privation, or want of knowledge, and it is absolutely false to say that the person who is deprived of knowledge must know oneself to be so deprived. That is contradiction at its most fundamental level. This is the common mistake of all such arguments, to classify doubt as a type of certainty, and insist that the uncertain person must be certain of one's uncertainty. But that is nonsense which completely mischaracterizes, and demonstrates a complete misunderstanding of uncertainty.

To properly understand uncertainty and doubt, we can look at the behaviour of children, babies, and other creatures which have not developed the certainty of epistemic knowledge. This gives us a far better understanding of the uncertainty which ought to be associated with radical skepticism, and radical doubt. The issue is with the understanding, or interpretation of meaning. If a person is unsure of what a word or combination of words means, that person will have real doubt with respect to the proposition. And if the person is uncertain of the meaning of all words, then that person has doubt about everything which is expressed in words.

So we cannot portray radical doubt, (or foundational uncertainty as I prefer to call it, in opposition to those who insist on a form of foundational certainty) in the way that Augustine does here. A person does not need to know oneself to be uncertain (doubtful) in order to be uncertain, and this is a very important fact about the nature of knowledge, which Socrates did an excellent job of demonstrating. Knowing-how is prior to knowing-that. So Socrates went through the entire range of human activities, from artists to craftspeople, to manufacturers, to mathematicians, scientists, lawyers, and theologians, and demonstrated that in all forms of human activity people were doing things who could not explain, or understand what they were doing. Doing precedes the human capacity to understand what one is doing. Consider as examples, a child learning to talk, or at the other extreme end of knowledge, the activities of quantum physicists.

The fact that we learn how to do something before we learn what we are doing, demonstrates the priority of uncertainty, because knowing-that involves a higher degree of certainty than knowing-how. And, it shows that it is not required that the person who doubts, knows oneself to be doubting. To portray uncertainty in this way, such that the person who is uncertain must be certain that oneself is uncertain, is a mistaken and profoundly incorrect representation of the nature of knowledge. Contrary to popular belief, knowledge is not based in certainty, it is supported by uncertainty. As Socrates said, wonder is the base of philosophy. And we know that wonder is a form of uncertainty. and philosophy is what supports knowledge. So we can conclude that knowledge is itself supported by uncertainty, the support being distinct from that which is supported.

I don't see how infinity is relevant. Your proposed "infinity of doubts" is unwarranted, and a person's consciousness may simply be a doubtful, or uncertain consciousness, i.e. the skeptic.

If you're doubting everything, of course you'd be doubtful of your doubting. Where's the problem?

He means you have to be certain of something, because when you TRY to doubt EVERYTHING then you can't doubt anything — Gregory

I don't see the logic. Why not just doubt everything? There's no problem with that.

:100:

As Kant said about the noumenal world (which is the same as the mind-independent world), nothing can be said about its objects except that they exist.

But when we say of the independent objects, that they exist, we use "exist" as a property. And, we've already said, that objects are a property of "the independent world". So we have already a double layer of predication. From this perspective, we've made the predicate "objects", into a subject and we proceed with another layer of predication, concerning the objects. But we are restricted in what we can say about "the objects" by the first premise, already assumed, that these are property of a further object, the world.

The issue is that the first object "the independent world", as a united thing, is not a justifiable assumption or conclusion. This assumption cannot be made as a premise as it is not at all supported by empirical observations, and we do not have the premises required to conclude logically that there is a united object which we can call "the independent world". So there is absolutely no support for "the independent world".

By Occam's razor, we cut this first principle, "the independent world", and we start with what sensation and intuition gives us, individual objects. In Platonic terms, the first principle is "the Many" instead of "the One", then we can proceed to enquire as to what makes an individual object an individual object, in a way which was exemplified by Aristotle, and from this enquiry we can derive principles to justify the existence of "the One", as an individual, rather than as a whole.

In this way we can proceed toward an understanding of what it means to "exist". The quoted passage says that all we can say of objects is that "they exist", but this is rather insignificant if we do not know the conditions for existence. So we dismiss the united object, whatever you call it, "the independent word", or "the universe", as an unsound premise which will only mislead, and ask what does it mean to be one of the many independent objects, thus justifying the concept of "existence".

Wittgenstein in his book On Certainty: "If you tried to doubt everything you would not get as far as doubting anything. The game of doubting itself presupposes certainty." — Gregory

This is incorrect, and Wittgenstein was obviously wrong with this principle. In reality, we go ahead and act when we are still in doubt of the outcome. Certainty is clearly not a requirement for acting.

So for example, earlier in this thread you gave Berkeley's reasoning for why objects extended in space can't exist, because of infinite divisibility. This ignores what modern physics has to say about subatomic particles and limits to length. It probably also ignores the resolution to Zeno's paradox in math. — Marchesk

Mathematicians do not have a resolution to Zeno's paradoxes, they just use a work around. And what modern physics says about subatomic particles, is to some extent a product of that work around, because physicists use it. But of course, modern physics is incapable of providing an understanding of subatomic particles.

So we ought to conclude that the work around which mathematics uses, is not very good.

There's an entire essay in this question, but to answer very briefly - I think 'eternal' is oversold for Platonic ideas and the like. It's more that they're non-temporal - that they don't come into or go out of existence - they're not temporally delimited or composed of parts. — Wayfarer

Temporality is often associated with change. If something changes, it is temporal, and if it is temporal, it must be generated and destroyed. These are the ideas which Aristotle grappled with in his physics, spatial change and temporality. So Aristotle would argue that if something is non-temporal, it must be never changing and therefore eternal. This is why "eternal", in the sense of Christian theology, has the meaning of outside of time, non-temporal, never changing, while "eternal" in the materialist or physicalist sense means endless time. You can see how different the two senses of "eternal" are, the theological one requires that time has limits, in order that there can be something outside of time, non-temporal, while the materialist one denies that time is limited.

When we posit the reality of something outside of time, "eternal", we must allow for the means for a relationship between the eternal, that which is outside time, and the temporal existents. Are you familiar with Aquinas' conception of "aeviternal", or "aeviternity"? He uses this conception to describe the medium, "the aevum", which is between the eternal and the temporal, and assigns to it the existence of the angels. I believe it's based in a Platonic proposal, of something which is changed, or changes when coming into being, but then is changeless.

The evolution of h. sapiens is fairly well understood. But I share with Alfred Russel Wallace scepticism that the intellectual, artistic and creative faculties can be understood solely through the lens of evolutionary biology. — Wayfarer

I don't think that the artistic and creative faculties of any living being can be understood through the lens of evolutionary biology.

How would you separate the aspects of human thought which are innate and eternal, as platonic realism dictates, from the aspects which are constructed by the human mind, and are "evolving"? Is this an example of the distinction between content and form? Would you propose that the content, being some fundamental platonic forms, which constitutes the subject matter of thinking, is distinguishable from the formal structures which the mind creates, as a "world-picture" for example.

The reason I ask this, is that you seem to adhere strongly to platonic realism, which would understand "ideas" or "Forms" as eternal unchanging, innate features of our intelligible universe, yet you also allow features which are constructions of the mind when you want to discredit naive realism. The following is a quote from a few posts back.

But it is interesting how both support the model of the mind as a constructive process that creates, generates or builds our world-picture, which seems to me to irrevocably disrupt the view of naive realism. — Wayfarer

The issue here, is the nominalist/realist debate which you often refer to. You seem to employ nominalist principles when you are arguing against some forms of realism, But then when it comes to supporting platonic realism, you appear to be anti-nominalist.

Would you be proposing some sort of hylomorphism of intelligible objects? In this case, a conceptual structure would consist of some parts which are eternal unchanging platonic ideals (the subject-matter or content of the intelligible object), and some parts would be constructs, produced or created by the human mind (the formal aspect of the intelligible object).

From this perspective, would we as human beings, have a vantage point, toward understanding the nature of true, pure, separate, independent, and immaterial Forms? If this form of dualism which you seem to be proposing places the innate, eternal Ideas, of platonic realism, as the subject matter, being the material content of the intelligible object, how can we turn this around to give true separate, existence to the independent Forms, as immaterial?

Do you see the point I'm making? This type of thinking, which gives priority to platonic realism, instead of denying all forms of realism, as a first principle of Socratic skepticism, gives us an upside down, or backward starting point. If instead, we assign "intelligibility" only to what is created by the mind, under Plato's principle of "the good", then we have a true starting point, to see that anything intelligible is necessarily created by a mind. And the idea of eternal, unchanging intelligible objects, as platonic forms, must be dismissed as incoherent.

There are distinct types of types. For example, there are types of qualities, colours, tastes, etc., and there are also types of objects, houses, cars, cups etc.. As universals employed in logic, the qualities are named as predicates, and the things are named as subjects. There is a big difference in role, between a type of thing, and a type of quality.

In the case of numbers however, there is ambiguity. Some would say that a number is a type of thing, and others would say that a number is a type of quality. And of course, it depends on how you use numbers, as subjects or as predicates. Is there a thing called "three", or is "three" a specific quality of a group?

But a difficulty emerges as soon as we ask ourselves how we know that a thing is white or a triangle. If we wish to avoid the universals whiteness and triangularity, we shall choose some particular patch of white or some particular triangle, and say that anything is white or a triangle if it has the right sort of resemblance to our chosen particular. But then the resemblance required will have to be a universal. Since there are many white things, the resemblance must hold between many pairs of particular white things; and this is the characteristic of a universal. It will be useless to say that there is a different resemblance for each pair, for then we shall have to say that these resemblances resemble each other, and thus at last we shall be forced to admit resemblance as a universal. The relation of resemblance, therefore, must be a true universal. And having been forced to admit this universal, we find that it is no longer worth while to invent difficult and unplausible theories to avoid the admission of such universals as whiteness and triangularity. ... — Bertrand Russell, World of Universals

I don't think that this is a correct portrayal. We do not utilize "resemblance", when judging distinct things as having the same quality. I think this is what Banno refers to when he mentions the use of "grey". Two distinct grey things, are not both judged as grey because they have some resemblance to each other. Instead, I think that such judgements ought to be seen as a sort of categorization.

So each grey thing is placed into the category of having the colour, or quality, of being grey. This is not a resemblance, which is a type of similarity, but it is a sort of sameness. And "same" is a very distinct category from "similar" because "similar" implies necessarily a difference, whereas "same" is to deny difference.

However, by the law of identity, we cannot say that the two distinct things, which are said to have the same quality, are the same in any unqualified or absolute way. So to facilitate communication, and show that we recognize the difference between them, it is commonly said that the two things are similar. But this common way of speaking veils the reality of the mental process which is behind this use of the same word to describe two distinct things. It misleads us into thinking that the two different things have been judged as being similar, when in reality they have been judged as being in some way the same, i.e. having the same type of quality.

And, that this is a case of being misled is justified by the fact that "same" is logically distinct from "similar", such that two distinct things having been judged as having the same type of quality does not necessitate that the two things are similar, unless we define "similar" in this way. But then we see that what constitutes "a type" is simply a definition, and we are led toward nominalism instead of the realism which you prefer.

There's an infinite number of numbers which no human has ever thought of. What's the point in trying to name a random one of these? Here's one for you though, which might be worthwhile. Try naming pi to its final decimal place. That's a meaningful number which no one has ever thought of.