I notice a deplorable fault in this forum. — InternetStranger

Mainly, I presume, on the part of myself.

The mathematical unit is thought from the Pythagoreans on as mental object, it has no external existence. — InternetStranger

That is a point I have often argued on forums, usually to complete incomprehension. In fact, my first post on any forum made this very point, back in 2009:

consider number. Obviously we all concur on what a number is, and mathematics is lawful; in other words, we can't just make up our own laws of numbers. But numbers don't 'exist' in the same sense that objects of perception do; there is no object called 'seven'. You might point at the numeral, 7, but that is just a symbol. What we concur on is a number of objects, but the number cannot be said to exist independent of its apprehension, at least, not in the same way objects apparently do. In what realm or sphere do numbers exist? 'Where' are numbers? Surely in the intellectual realm, of which perception is an irreducible part. So numbers are not 'objective' in the same way that 'things' are.

Interested in your comments on that.

However,

There was no applied math for Galileo. he made an ontological claim, about the substance of the universe totality, based on a thought experiment. — InternetStranger

I can't understand how this could be true, as so much of Galileo's fame rests on precise measurement. How can measurement of celestial objects or mass not depend on applied mathematics?

Mainly, I presume, on the part of myself. — Wayfarer

Certainly, congratulations, but I think it is, also, more perfectly general, almost a universal disease.

I am prolix, due to some clarity, ergo, I try to see this all more clearly through the survey which does not yet acquire the full masterly pinnacle of vantage over the material in question.

“that we concur on is a number of objects, but the number cannot be said to exist independent of its apprehension, at least, not in the same way objects apparently do.”

This is not the same issue. That is the current way of using numbers. What the Greeks say is: there is no thing that corresponds to the number. Counting numbers are not units. Units are only in the mind. There is not one apple, since it is a different object in the region of change, than another thing that is the same. Same doesn’t mean the same thing as equal. Even if the atoms of the whole apple were of the same counting number. Same can mean, belongs together, and it can mean somehow an “identity”, though, strictly speaking, the only identity is with an apple and itself. Another shares in the same form or idea, or genus.

The unclarity here lies in the radical difference the ancients gave to heavenly motion, and earthly. And its transitional phase in Galileo, with half a foot in the medieval, "astronomy is math in time and place". And another moving towards the modern view.

“So numbers are not 'objective' in the same way that 'things' are.”

The Greeks don’t have a notion of objective as what is not dependant on the human being. What they mean is that the intellect is literally a manner of perception. It is ontological. Speaks of the substance. Modern physicists, apart from a few like Penrose, don’t regard maths that way.


“I can't understand how this could be true, as so much of Galileo's fame rests on precise measurement. How can measurement of celestial objects or mass not depend on applied mathematics?”

Because they regard it as a claim about what the appearances are. The truth about the phenomena. Hypothesis used to mean throwing something under, into the substance. Notice the prefix, hypo. A putting under in the supposatium or foundation. The word hypothesis has changed meaning, now it means a working methodological consideration that wants to bring forward a result. It works! Success! that’s why I used the Rutherford example above, to point to this change. There was no, from a doctrinal point of view, I don’t speak de facto, applied mathematics in former times.

Measurements are tools in our (modern) usage, which is why we have maths of approximation. It’s simply a gauge of the rigour of the science so far as it can bring out precice results. It’s linked to practice, not to ontological knowledge. Remember the word of Feynman, rigour in the sciences is the number of decimal points. He says, also, of concepts in physics, it must do something, only then do we know it exists.

Interesting perspective, thanks. Some rather strange turns of phrase:

I try to see this all more clearly through the survey which does not yet acquire the full masterly pinnacle of vantage over the material in question. — InternetStranger

But I nevertheless think I understand what you're driving at.

What the Greeks say is: there is no thing that corresponds to the number. Counting numbers are not units. Units are only in the mind. There is not "one apple", since it is a different object in the region of change, than another thing that is the same. — InternetStranger

Right - I think I understand that. I happened upon an encyclopedia entry about a German neo-Kantian philosopher by the name of Afrikan Spir which has this to say:

For Spir the principle of identity is not only the fundamental law of knowledge, it is also an ontological principle, expression of the unconditioned essence of reality (Realität=Identität mit sich), which is opposed to the empirical reality (Wirklichkeit), which in turn is evolution (Geschehen). The principle of identity displays the essence of reality: only that which is identical to itself is real, the empirical world is ever-changing, therefore it is not real. Thus the empirical world has an illusory character, because phenomena are ever-changing, and empirical reality is unknowable.

Similar principle, I believe.

The Greeks don’t have a notion of objective as what is not dependent on the human being. What they mean is that the intellect is literally a manner of perception. It is ontological. — InternetStranger

I think I see what you mean here also. And also the point you make about 'whatever works' - that really is another way of describing the so-called 'falsifiability principle'.

I would be interested if you had a look at this passage in the Cambridge Companion to Augustine. I have often referred to it over the years on these forums, but nobody seems to get what he's talking about.

“Afrikan Spir”

It’s the Laws of Thought/”logic” in Aristotle's Metaphysics. Law of identity. Human reality as speach, i.e., the human essence. Because otherwise there is radical individuality, even inexpressible under the generic term individual. Ousia as the ground for all change. It’s confusing because the medievals talk of substance and hypostasis (ousia) as the “object” of change.

Not sure this states anything unusual or particular to Spir. Beside for the emphasis on that principle over the others.


“ 'whatever works' - that really is another way of describing the so-called 'falsifiability principle'.”

I give that as corresponding to the vernacular or general public notion of Science. Science does stuff. It is not the same as Popper’s methodological idea, he says a thing tested can’t conform to all possible observable data. Something must falsify the claim. Science doesn’t mainly operate in Popper’s world, because scientists don’t largely care about proving things. They collect observations about how things happen. Anything that happens, whether or not it proves anything, is scientific as strict observation.



Augustine.

You’re right, it’s object of the intellect, not of the senses. not sure how far the Greeks bring that out. They speak of the intellect as reaching the true world and the true field and the true sky, the eternal things, as being the thing that may be immortal, the intellect is a angel in the theological modification. For us the objective is for the senses. Independent external reality as mathematically certain was the idea from Galileo & especially brought forth in Descartes and questioned in Newton, but gradually it becomes “fact” as thing testable, as in the discussion between Hobbes and Boyle and the Royal Society.

When we say objective we mean independent of humans in the sense that people say the stuff happening billions of years ago didn’t require humans. If one closes one’s eyes, the world is still there. Not dependant on humans, objective. This is rather vague in current discussions. Ordinary sense, not math, is objectified as “facts”. Newton said he made no hypothesis because he held his view of gravity to be based on the senses. Not an envisaging of these internal objects. The maths he could understand as external rules imposed on the heavens, which were no longer the realm of mathematical circular motion differing from earthly motion. The word hypothesis lost its meaning. newton claimed to have a “doctrine”, but those used to using the word hypothesis thrust it back at him and he eventually gave way.

This is taking me out of my depth, as you see; I need to study these things more deeply.

Units are only in the mind. — InternetStranger

But that is the mistake that leads to strong Platonism. So it would be the other deplorable fault that pervades this forum. :)

The interesting thing about the world is that aspect that maths captures - the fact that forms constrain material being. Individuation is contextual. The identities of things are the result of differences being suppressed to the point they cease to make a difference.

So counting is based on that trick. A mathematical unit, like 1 or 0, is defined by an identity operation - the transformations which don't actually change it. Multiplying by 1 changes nothing. Adding 0 again changes nothing. A unit speaks to a state of perfect symmetry. And having constrained change so as to arrive at an unchangeable symmetry - the unit - then something new can happen. Construction can begin employing that now stable bit of identity. Constraint produces the very thing which is its antithesis. The freedom to start building up from definite parts.

So units are really out there in the world. The Platonic forms are descriptions of constrained symmetries. And to the degree material instability is thus regulated, atomistic construction can begin. We have individuated individuals - like electrons or other identical quantum particles that lack any essential differences ... and so now only have the antithetical thing of particular contextual properties, like how they break the global symmetries of space, time and energy.

To talk of units being only in the mind is to yield to substance dualism. Instead, the possibility of units is already inherent in the world because formal constraints have limits.

The rough and irregular edges can be rubbed off any object. But eventually that in turn means one is going to arrive at the smooth and the regular - the most symmetrical individual possible. And so the further possibility of unit-based, atomistic, construction is already anticipated in that smoothing process. It is immanent in reality that arithmetical operations like adding and multiplying will be present to the degree that individuation has been most fully realised.

So no need to invoke Platonic realms or the power of human minds. Units will emerge naturally where constraint is allowed to act to suppress difference. In trying to erase something - individuation - reality must in fact end up creating it in the guise of something new, the atomistic ability to construct.

"But that is the mistake that leads to strong Platonism. So it would be the other deplorable fault that pervades this forum. "

Sure, but for readers of history of philosophy as readers of that history, it is surely not wanted. Since they want to see things the way the thinkers did, think in and through their ruts truly and properly.


I don't think one form has anything mathematical to do with one unit in the Greek sense. As idea it is the genus of "one". But what does one mean, a whole. A form is a whole, one man, cut an arm off, no longer whole. There is no mathematical equality between men. They are the same as they are under the same form or genus. They are both one is the everyday sense. That's not mathematical in the Greek sense. It's practical everyday vague, not exact, counting.

" It is immanent in reality that arithmetical operations like adding and multiplying will be present to the degree that individuation has been most fully realised."

In ordinary life we can't jump to five million. In maths we don't have to wait to count 1,2,,4, etc., we go right where we want. That's wholly unlike life. Infinity is intelligible, I count, 1,2,3, well, it goes on I say to myself tacitly, as it were, infinity. No such thing in the world as what one can point to.

"Units will emerge naturally"

Not sure how any natural things, if that means stuff one can point to, can be equal in the perfect sense things are equal in the mind. e.g., an angle of 90 degrees. never occurs for the senses by the Greek way of thinking. I don't think form is like number, in fact, number in the mathematical sense of unit is a form. I.e., it is something peculiar, unlike anything else.

When we say "objective" we mean independent of humans in the sense that people say the stuff happening billions of years ago didn’t require humans. If one closes one’s eyes, the world is still there. Not dependant on humans, objective. This is rather vague in current discussions. — InternetStranger

It's vague because it's assumed. It is the attitude distinctively characteristic of the advent of the modern period and the rejection of scholastic and traditional metaphysics.

But the human mind is always and inextricably involved in whatever is discovered by science, as it is the mind that introduces perspective, scale, and duration, even if only implicitly. But then modern science brackets out the subject and claims, therefore, to arrive at a 'view from nowhere', the universe 'as it really is' were nobody to be observing it. //ps Hence Schopenhauer's observation that 'materialism is the philosophy of the subject that has forgotten itself' or something to that effect.//

This is taking me out of my depth, as you see; I need to study these things more deeply. — InternetStranger

Well, it's a deep subject, and thanks for your reflections on it.

(The book I mentioned, E A Burtt, The Metaphysical Foundations of Modern Science, really does go into these questions. It is a standard text in history and philosophy of science, published in the 1930's. I acquired a copy several years ago, having been told about it on these forums, but it's a difficult book and didn't get far into it, must make another effort.)

I don't think one form has anything mathematical to do with one unit in the Greek sense. As idea it is the genus of "one". But what does one mean, a whole. A form is a whole, one man, cut an arm off, no longer whole. There is no mathematical equality between men. They are the same as they are under the same form or genus. They are both one is the everyday sense. That's not mathematical in the Greek sense. It's practical everyday vague, not exact, counting. — InternetStranger

Not so. We can talk about the unit triangle just like the unit one. The most individuated possible things are precisely those that are the most symmetrical versions imaginable.

But now you are talking about a world of hierarchical complexity - where both top-down constraint and bottom-up construction are in play. So there is something it is like - a genus-level constraint - that it is to be a human. And there are then the particular historical accidents that also compose that human.

A man might have lost an arm - but he was meant to have one. The genetic intent existed. That constraint on growth was there. It was simply a historical contingency that it ran into a chainsaw.

And a man might be bald, shaven or hirsute. That again is some kind of accident in regards to what we consider as a necessary constraint flowing from the genus. It's optional because it is a difference that doesn't make a difference. And so the genus "human" has the kind of generalised symmetry I'm talking about.

Maths just takes that way of thinking and imagines it with all "material accidents" or "historical contingencies" shorn away to leave a bare formal necessity. That certainly works as act of imagining what a perfect limit would look like. But we can see the trick of the imagination that is involved in turning an immanent development towards an ultimate limit into some transcendent claim that the limit exists in some Platonically dualistic realm.

In ordinary life we can't jump to five million. In maths we don't have to wait to count 1,2,,4, etc., we go right where we want. That's wholly unlike life. Infinity is intelligible, I count, 1,2,3, well, it goes on I say to myself tacitly, as it were, infinity. No such thing in the world as what one can point to. — InternetStranger

Yeah. Construction allows that kind of freedom, as I say. It is the very opposite of constraint, even if it is ultimately the product of that constraint.

Not sure how any natural things, if that means stuff one can point to, can be equal in the perfect sense things are equal in the mind. e.g., an angle of 90 degrees. never occurs for the senses by the Greek way of thinking. I don't think form is like number, in fact, number in the mathematical sense of unit is a form. I.e., it is something peculiar, unlike anything else. — InternetStranger

Your objection here is not clear so I can't answer.

“Not so. We can talk about the unit triangle just like the unit one. The most individuated possible things are precisely those that are the most symmetrical versions imaginable.”

I was speaking about the Greek conception of mathematics, so “not so” is not an appropriate answer to be followed by a claim to propound the truth, in contradistinction to the Greek view, about the issue. I think, also, your truth doesn't make sense. Things, e.g., monads, can never be equal, how could they be considering equal means not different, but, rather, perfectly the same. A performative contradiction.


“Maths just takes that way of thinking and imagines it with all "material accidents" or "historical contingencies"

Sure, but this is applied maths. Like maths of approximations. It would be useless without the logical connection which depends on consistency across the terms, function means translatability of the terms. It doesn’t work with genuine idiosyncratic form, only with randomized pseudo-uniqueness. It’s true so far as one defines reality as what maths can treat in that manner. Than the question is, how far is this definition of reality justified?

“Construction allows that kind of freedom”

What is “construction”? That makes no sense, I’m speaking empirically. Maybe for a super-being.

“Your objection here is not clear so I can't answer.”

We have the idea that two right angles are equal to all the angles of a triangle. It never matches exactly with existing triangles. This is as true for the Greeks as that we have the idea of a triangle, but a real triangle is always qualified, it is, e.g., isosceles.

All this corresponds to the truth that math is a form, it needs form for its foundation. Math is a qualified form of the form of form. The form of form is the only perfect or unqualified form.

“ But then modern science brackets out the subject and claims, therefore, to arrive at a 'view from nowhere', the universe 'as it really is' were nobody to be observing it.”

This is vexed. Because the Positivist view would not, then, be vague. It’s exact. Reality, and, correspondingly scientific description, is given to the decimal point, it is what is quantifiable. The ordinary vague sense of the fact, by contrast, corresponds to the sense of fact established in the debate between Boyle and Hobbes, and promulgated through the Royal Society. It corresponds to testing, describing things as repeatable, Established facts. The pre-Royal Society meaning of fact was the legal term, “accessory after the fact”, it meant knowing or legally culpable act, the early 18th century still speaks of, e.g., “ignoble facts”, meaning depraved actions. The old fact meant the human action rather than natural occurrence. Blind nature and fact were opposed. Ergo, there is a deep cauldron of confusion corresponding to the general existentialism of the public. Whereas, in the university, there are hard or scientific facts that are not values or ideologies.

Ergo, science is done by existentialists, which, when doing science, are Positivists.

I was speaking about the Greek conception of mathematics — InternetStranger

It is not exactly clear what you think that conception actually was.

Are you saying it relied on the concept of the unit more than of deductive proof?

Things, e.g., monads, can never be equal, how could they be considering equal means not different, but, rather, perfectly the same. A performative contradiction. — InternetStranger

Things are the same to the degree there are no differences that count. What's the problem there?

No. It's more simple. Math is not for the senses strictly speaking. I'm saying just as when I count one, two, three, cups in the cabinet, I don't do math, since I don't deal in strict logical relations, 1 + 1 + 1 = 3, i.e., 3 units, available only to the mind, dianoia or episteme. No two cups are exactly equal. No triangle or geometric form is part of experience, neither anything done with them. Isn't it obvious that a rough drawing in dirt of a square is not a true and proper square? Neither any relations of angles, areas, and so on.

"Things are the same to the degree there are no differences that count. What's the problem there?"

Depends on what you want to do. Greeks were aiming at true knowledge of the cosmos, not building technology. Besides, it's not obvious there is anything close to reality there, I don't believe in thirty thousand years anything physicists now think is a good unified theory, or a Plank limit, or such things will hold up. Not even be taken seriously.

Reality, and, correspondingly scientific description, is given to the decimal point, it is what is quantifiable. — InternetStranger

It is 'given' only in respect of those aspects of reality which are amenable to quantitative measurement; which is a key point. For Galileo, and those afterwards, the 'primary qualities' were just those attributes which could be thus measured, and the mind of the observing subject implicitly located amongst the 'secondary qualities'. That is one of the crucial differences between modern and pre-modern epistemology and has had considerable consequences.

I take it that what Internet Stranger is referring to, is the Platonic or generally traditionalist view of the intelligibility of number (etc.) What I've been able to learn about this topic is that, for the traditionalist, the knowledge of numbers (etc) was different in kind to the knowledge of sensible particulars (i.e. objects as such), because such knowledge was immediate and apodictic. This means that in the knowledge of forms, ideas, geometric proofs and the like, the truth is apparent to 'the mind's eye' in a manner that is not possible with the knowledge of sensible objects. That is my interpretation of what is being referred to here; I'm not sure that I understand it very well, but I'm pretty sure (with all due respect) that you're not understanding it at all. And that's because you're viewing it from your modern/system science/biosemiotic perspective, rather than from the 'traditionalist' perspective. So here you need to be aware of the spectacles you're looking at the question through, so to speak.

One of my 'canned quotations' is apposite to this point; the passage in Lloyd Gerson, 'Platonism vs Naturalism', where Gerson is speaking about Aristotelian epistemology:

in thinking, the intelligible object or form is present in the intellect, and thinking itself is the identification of the intellect with this intelligible. Among other things, this means that you could not think if materialism is true… . Thinking is not something that is, in principle, like sensing or perceiving; this is because thinking is a universalising activity. This is what this means: when you think, you see - mentally see - a form which could not, in principle, be identical with a particular - including a particular neurological element, a circuit, or a state of a circuit, or a synapse, and so on. This is so because the object of thinking is universal, or the mind is operating universally.

….the fact that in thinking, your mind is identical with the form that it thinks, means (for Aristotle and for all Platonists) that since the form 'thought' is detached from matter, 'mind' is immaterial too.

Now, of course, I don't for a minute expect that will accept that, as in your ontology, there is no provision for anything immaterial and because of the obvious implied dualism. However this is what needs to be stated in order to understand what is actually being said, in my opinion.

(Actually I had drafted the above before Internet Stranger entered the post above this one, but I think there's a fair amount of agreement.)

“Only in respect of those aspects of reality which are amenable to quantitative measurement;”

That's simply not the scientific conception of reality. Reality is the measurable. What counts as foundation is what is measurable. Physics is thought not under a Lockean scheme, which is, I admit, like the “two tables” business, but which also makes no sense, since touch or solidity is also a subjective sense, people with no sense of solidity or pain, the two of which coincide, exist, just as do the blind. Though they often die young. No, Plank shows the foundation, the limit of quantifiability. And the rest is thought as emergent (notably in Comte), or, as sheer illusion. Ergo, one hears often, in the tradition stemming from Hume, of “folk” understanding. The folk world and the illusion of consciousness.

In other words, science in the strict sense deals with the measurable. The rest is unscientific.Science deals with reality, the rest is fiction. Hume goes so far to say this, as you will know, of causality. This is, parentheticaly, how the misuse of the notion of a genetic fallacy has come into the west. Since Kant granted that we could not say causality is false because of its limited or dubitative source in the human psychology, and the Logical Positivists took this up and started justifying everything out of it. Since, the alternative is to admit human judgment can’t be the foundation for logic and science.

You've attributed something I (Wayfarer) said, to Apokrisis - or that is how it comes across.

This means that in the knowledge of forms, ideas, geometric proofs and the like, the truth is apparent to 'the mind's eye' in a manner that is not possible with the knowledge of sensible objects. — Wayfarer

But that is my point. Once you realise that the world has accidental particulars, that is how you start to discover its necessary universals.

So it is categorising reality in terms of the one that also reveals the reality of the other. It is because we can conceptualise an aspect of every actual substantial thing as being the result of a material accident that we are also justified in dialectical fashion to conceptualise everything that is not a material accident to be a formal necessity.

You make your own jump from that hylomorphic view of substantial actuality to a Platonic story about the mind having some mystic access to another kind of reality. One gains access to a transcendent realm of pure forms by leaving behind the dirty, dusty, imperfect world of the actual.

But my position is the Aristotelian one. Matter and form are the two aspects of substantial being - one standing for constructive causation, the other for the causality of constraints. And both are immanent of this actual world.

So the way we "see" the realm of form is via the shedding of the accidental particulars. We keep rubbing away every unnecessary rough corner. And eventually we find the symmetry, the limit, where only the necessary formal structure remains.

You are claiming this is an exercise in transcendent perception. But it is just a pragmatic exercise of getting rid of the "obscuring" details. It is an empirical story as much as a rational story as we have to work out what material facts can be ignored.

Now, of course, I don't for a minute expect that will accept that, as in your ontology, there is no provision for anything immaterial and because of the obvious implied dualism. — Wayfarer

It is not that I don't make a provision for anything immaterial. Mine is not a sin of omission.

What I am arguing is that it is as plain as the nose on your face that abstraction or generalisation proceeds by discovering what empirical facts you can afford to ignore. You will arrive at the bare forms of things, the necessary structures, apophatically. It is what is left once everything that can be left out has been left out.

So it is the feature of my ontology that it is hylomorphic and not Platonic. It is immanent and not transcendent. It is a process view and not an eternalist one. It sees flux and development as basic, not stasis and existence.

And that's because you're viewing it from your modern/system science/biosemiotic perspective, rather than from the 'traditionalist' perspective. — Wayfarer

Well it should be clear enough that I am view it from a natural philosophy angle that goes back to Aristotle and Anaximander as opposed to some theistic framework like yours.

But you are doing your thing - trying to pigeon hole everything I say as Scientism at work. :roll:

But you are doing your thing - trying to pigeon hole everything I say as Scientism at work. :roll: — apokrisis

I'm really not. When I said 'with all due respect', I meant it. And I still think you're not seeing the point. You're re-interpreting the whole question from a modernist perspective. Furthermore, the Platonic perspective that I mentioned is not 'my invention' - it's not something I thought up. But, you see, you regard it as a virtue to 'omit the eternal' - that's what you regard as 'de-mystifying telos', I think was the phrase - whereas for the traditionalist, the disclosure of the eternal was all that really mattered. And that's not even talking about anything that I do or don't believe - it is purely a question of hermeneutics, i.e. the interpretation of the tradition.

You're re-interpreting the whole question from a modernist perspective. ... But, you see, you regard it as a virtue to 'omit the eternal' — Wayfarer

Well sure. I'm not here to speak for the authentic 300BC Aristotle. In his day, there was far less reason to take a strong developmental approach to Cosmology - as science now requires us to do with the Big Bang. So he would have been more inclined to eternalism in his ontology.

So yes, we could have a more historical conversation about what was actually believed in terms of what was empirically known at some stage of the development of these contrasting traditions. But it is silly to say that I omit the eternal as "a virtue". I am responding to what we have learnt.

The "virtue" here - the scholarly one that I do value - is giving full credit to the history of interesting ideas. If Aristotelian tradition was by now a completely dead one - as Enlightenment science tried to proclaim - then I wouldn't even bother to mention it. But I stress it precisely because it is still relevant and influential ... for systems scientists and natural philosophy.

If Aristotelian tradition was by now a completely dead one - as Enlightenment science tried to proclaim - then I wouldn't even bother to mention it. But I stress it precisely because it is still relevant and influential ... for systems scientists and natural philosophy. — apokrisis

Yes indeed, and a good thing too. You might find this review of interest.

Thanks. Looks on the money. :grin: