I have never used infinity as anything more than unboundedness. — jgill

In Calculus 1 classes, there is not a concern that the subject be axiomatized. But if we are concerned with having the subject axiomatized, then the ordinary mathematical context is one in which there are infinite sets. Just take an infinite sequence of real numbers. A sequence is a kind of function, and every function has a domain, and the domain of an infinite sequence is the infinite set of natural numbers.

The inverse relation between points and continua is that the point is nothing and the continuum is something. — keystone

You mean the continuum is everything. That is the opposite of nothing. Then what you call continua are the line segments that are fall inbetween these two complementary extremes.

In terms of length, the point is exactly 0 and the line is some positive number. If you're talking about something of infinitesimal length, you are talking about some tiny line segment. If that's the case then you are only talking about continua. — keystone

These are your words. But if the line is cut, then you are also talking about a lack of line with some infinitesimal length, not a 0D point.

This just helps show that the idea of a 0D point is ontically problematic and in need of much better motivation than you are providing. You assume too much without providing the workings-out.

Nothing cannot be used to measure something (0+0+0+0... always equals 0). Whereas something can be use to measure nothing (e.g. 5-5=0). There is an imbalance here in this relationship suggesting that continua are more fundamental. — keystone

Nothing and everything are really the same. A void and a plenum are either too empty to admit change, or too full to admit change. White noise is both every song ever written, or that even could be written, played all a once, and no song being played at all.

Continua are certainly then something. And since something cannot come from nothing, continua must exist as a constraint on a state of everything. This is the better route to getting towards the intuitionist view of the continuum as having numbers with as many decimal places as you care to produce them.

The constraint on mining the number line for some particular value at a point is time and energy. If you develop a tight enough context, you can produce a matching degree of certainty.

Again, it is all about the reciprocal relation. The one physics cashes out in terms of entropy and information these days.

The structure of a continuum is not defined by points, it is defined by an equation(s). Aside from being impossible, you would never try to provide an infinite list of points to completely describe a line (Cantor). You would just provide an equation - a finite string of characters to perfectly describe how points would emerge if cuts are made. — keystone

Sure. Behind it all is symmetry and symmetry breaking. Numbers are based on the maximum symmetry that is their identity operation - 0 for addition, 1 for multiplication. This first step suffices to produce the integers. Then more complex algebra gives you further levels of symmetry to populate the number line more densely with other symmetry breakings.

There are generators of the patterns. You start with the differences that don’t make a difference. Then this yields a definition of the differences that do.

Again the logic of the dialectic and the basis of semiotics. Stasis and flux are a dichotomy. Mutually dependent and jointly exhaustive. Each is the measure or the other.

I don't understand how continua + equations are vague. If I say I'm thinking of a plot containing the curves x=0, y=0, and y=x^2 you know exactly what I'm thinking of. — keystone

The operation is crisp or determinate to the degree it is robustly dichotomised. It is vague to the degree there could be some doubt.

To use the usual example, when you say x=0, are you talking about 0.00…. to some countable number of decimal places. Have you excluded x=0.0000….a gazillion places later …0001?

There could be a vagueness about this nought of which you speak so freely. There could be some uncertainty you have failed to eliminate.

The OP mentions Aristotle's distinction of actual vs. potential infinities. The Wikipedia page on the subject doesn't explain the difference between the two all that well. — Agent Smith

The adjective 'is infinite' is mathematically defined in the formal theory, set theory. I have seen no formal theory in which the adjective 'is potentially infinite' is mathematically defined.

A mathematical definition of an adjective 'P' is of the form:

Px <-> Q,

where 'Q' is a formula that has no free variables other than possibly 'x' and no symbols other than the primitives or previously defined symbols.

It's something to keep in mind that people who use 'is infinite' are at least backed up by a mathematical definition, while those who rely on a notion of 'potentially infinite' are not.

I find it hard to imagine that something (an n-dimensional continuum) can be constructed from nothing (0-dimensional points). — keystone

First, there are two different notions of 'the continuum'. One is that the continuum is the set of real numbers R. The other is more specifically that the continuum is R along with the standard ordering on R, or formally the ordered pair <R L> where L is the standard 'less than' ordering on R.

Second, where can one read of a notion of the real continuum as an "n-dimensional continuum"? What does it mean?

Third, the set of real numbers is not constructed from nothing. The set of real numbers is constructed as the set of Dedekind cuts of rational numbers (alternatively, as equivalence classes of Cauchy sequences of rational numbers), and the rational numbers are constructed as equivalence classes of integers, and the integers are constructed as equivalence classes of natural numbers, and the set of natural numbers is derived axiomatically from the set theory axioms. That is not nothing.

/

Suggestion: Since you are interested in formulating an alternative to infinitistic mathematics, then you would do yourself a favor by first reading how infinitistic mathematics is actually formulated, as opposed to how you only think it's formulated, and also you could read about non-infinitistic alternative formulations that have already been given by mathematicians.

/

I do feel that there is a little bit of Cantor's nonsense implied in any view that supports actual infinities. — keystone

That's what you feel. But you've not supported it. If by "Cantor's nonsense" you mean his religious beliefs, then it is plain, flat out false that axiomatic infinitistic mathematics implies Cantor's religious beliefs.

This is important to recognize:

(1) Cantor's work is not axiomatic. His work was from before the modern axiomatic method reached a satisfactory stage. There are mathematical difficulties with Cantor's work due to the fact that it's not axiomatic.

(2) No matter what one thinks of Cantor's religious beliefs and how he related them to mathematics, we can separate the wheat from the chaff by recognizing the conceptual advantages of Cantor's set theoretic work without also including his religious views about it.

(3) The inconsistencies that he tried to explain by religious notions do not (as far as we know) occur in actual axiomatizations by later mathematicians.

(4) Given (2), other than for historical appreciation and for informal motivation, it is not needed for mathematicians to refer to Cantor at all. Once we had Zermelo's work, and then as it itself is rendered by the methods of symbolic logic, for purposes of formal mathematics, we could forget that Cantor even existed.

I've heard people say that the paradoxes entwined with actual infinities are beautifully mysterious...I just think they demonstrate the flaws of the concept of actual infinity. — keystone

What specific paradoxes do you refer to?

Keep in mind that no contradiction has been found in ZFC.

Why can't we just say that pi is not a number? Instead, it is an algorithm — keystone

Fine. But it's not easy to axiomatize real analysis that way.

One can philosophize all day about how one thinks mathematics should be. But other folks will ask "What are your axioms?" They ask because they expect that an alternative mathematics should have the objectivity of set theory, which is utter objectivity in the sense that, by purely algorithmic means, we can definitively determine whether a purported proof is actually a proof.

A bit of magic is needed to make the leap from a finite collection of points forming nothing to an infinite collection of points forming a continuum. — keystone

As I mentioned, that is not how it is done. You would do yourself a favor by reading a good textbook on the subject so that you would have a basis to critique the actual mathematics rather than what you only imagine is the actual mathematics.

I think I have a grasp of how real numbers play into accepted set theory — keystone

You don't.

it is challenging for me to envision the existence of a set of all natural numbers. Without assuming its existence, accepted set theory doesn't get far off the ground. — keystone

The existence of the set of natural numbers is derived axiomatically. Granted, the key axiom is that there exists a successor inductive set, which is an infinitistic assumption.

On the other hand, the notion of "potential infinity" demands alternative axioms.

Take just the non-infinitistic axioms of set theory. What axioms does the "potential infinity" proponent add to get real analysis?

∞ isn't and object — Agent Smith

That's right.

In previous threads I've pointed out that in set theory there is not a noun 'infinity'.* Rather there is the adjective 'is infinite'.

This is a crucial point for understanding how the subject of infinite sets is approached in set theory. To continue to ignore this point is to commit to continual confusion about the very foundational notions.

* Notations such as 'as n goes to infinity' are abbreviations for formal writing that dispenses with 'infinity' as a noun. And points of infinity and negative infinity are something different too.

:up: And if you agree ... you would be a finitist, si señor?

No. You're just being glib and not even thinking about what I wrote; just pouncing in an ill considered way about it.

First, 'finitist' has many different senses in the philosophy of mathematics.

Second, recognizing that set theory doesn't have 'infinity' as a noun but rather 'is infinite' as an adjective, does not at all entail that one shouldn't also assert that there exist sets that are infinite.

I'll leave you to discuss with the other experts. Good day.

Cantor lost his mind (theia mania) and spent his later years in a lunatic asylum for instance. These concepts & paradoxes of which there are many seem to have a deletorious effect on the brain/mind - constantly mulling over them may lead to a nervous breakdown. — Agent Smith

(1) By what source do you assert that Cantor "lost his mind"?

Cantor had collapses and severe depression. I don't know of any source that says he "lost his mind" in the sense of insanity such as schizophrenia, delusions or hallucinations. One does not ordinarily say of people who are depressed that they "lost their mind".

(2) By what source do you assert that Cantor was in a "lunatic asylum" (thus suggesting that he was himself a lunatic)?

Cantor went to sanitoriums for his collapses and depression. I don't know of any source that says he was instutionalized as a lunatic.

(3) It is not a safe inference that Cantor's mathematical work itself caused his collapses. Famously there were other stressors at work.

/

Now, let's deal with the heart of this. Whatever mental problems Cantor had, they do not refute the insights of his work. Otherwise, would be an ad hominem argument. And I mentioned also that the core of his work is easily detachable from his religious beliefs.

It's fine to be interested in Cantor's biography, but it's beneath the dignity of even as common a forum as this to argue or even insinuate that his mental difficulties enter into a fair evaluation of his work.

I'll leave you to discuss with the other experts. Good day. — Agent Smith

Yeah, you often ditch an exchange with that arch "Good day" sign off, while not ingesting a single bit of the information and explanation given to you.

infinite object, something beyond our comprehension — keystone

I comprehend the notion of an infinite set.

logicism, the ideology that there is a single correct logical definition of a mathematical object — sime

That is not what logicism is.

I have nothing more to contribute. — Agent Smith

The word 'more' there is excess.

You raise some valid points in re Cantor's mental issues; maybe we need someone else, an arbiter, to sort out the matter between us, si señor tonesindeepfreeze. Medical records are usually confidential and what's released into the public domain maybe only bits & pieces of the whole story.

Yeah, you often sign out of a conversation with that snarky "Good day", while not ingesting a single bit of the information and explanation given to you. — TonesInDeepFreeze

Well, truth is you're too technical for my taste. Not your fault though!

The word 'more' there is excess. — TonesInDeepFreeze

:smile: You're the expert, you would know!

you would never try to provide an infinite list of points to completely describe a line (Cantor) — keystone

Cantor doesn't do that. In fact, Cantor proved that that CAN'T be done. It's his MOST famous result.

You have it completely backwards.

What articles have you read about Cantor that have led you to your terrible misunderstandings?

EDIT: I struck out my message here. I misunderstood the poster. He did not say that Cantor said the continuum can be listed; rather he said that Cantor said that that cannot be done, which is correct. My message is fully retracted.

maybe we need someone else, an arbiter — Agent Smith

Actually, you could start by just refraining from making claims for which you have no basis.

But if you do want to know more about Cantor's life then there is the Dauben biography.

you're too technical for my taste. — Agent Smith

That's such a cop out. When a person such as you posts a bunch of wildly intellectually irresponsible, incorrect and confused bull, it's not being "too technical" for me to flag it and sometimes even, gratis, provide explanation regarding it.

Anyway, I am truly curious why you thought of extrapolating from the fact that Cantor had breakdowns and depressions and was in sanitoriums to claiming that he "lost his mind" and was in a "lunatic asylum". Or you just like to bolster your point of view about mathematics by making stuff up and post it as if it's fact?

Damn. — john27

:rofl:

I have never used infinity as anything more than unboundedness. — jgill

In Calculus 1 classes, there is not a concern that the subject be axiomatized. But if we are concerned with having the subject axiomatized, then the ordinary mathematical context is one in which there are infinite sets. — TonesInDeepFreeze

I'm not sure how your post relates to my quote. To clarify, I've taught both undergraduate and graduate courses in (mathematical) analysis and published papers in the subject, and I have only very rarely had to resort to a transfinite argument or even read such an argument. In fact, the only time I can recall is the Hahn-Banach theorem in the functional analysis grad course I took many years ago. The proof involves the Hausdorff Maximal Principle (i.e. Axiom of Choice or Zorn's Lemma) and even there if one strengthens the hypotheses just a tad HMP or AOC or ZL can be avoided.

All math majors learn a little naive set theory and some ZFC these days, so mathematicians and students work in an environment underlaid by the fundamentals of the real line. It's just that conversations involving cardinalities beyond ${{\aleph }_{1}}$ don't usually occur in classical or even much of modern analysis.

Wiki says this:

The aleph numbers differ from the infinity ( ∞ {\displaystyle \,\infty \,} {\displaystyle \,\infty \,}) commonly found in algebra and calculus, in that the alephs measure the sizes of sets, while infinity is commonly defined either as an extreme limit of the real number line (applied to a function or sequence that "diverges to infinity" or "increases without bound"), or as an extreme point of the extended real number line.

I don't want to dive into what happens at the Planck scale, — keystone

If you want to argue for potential infinities over actual infinities, then the real world is surely the better place to test your case.

Arguing against maths using physicalist intuition becomes Quixotic if maths simply doesn’t care about such things. Physics at least cares.

What I have said is that - as the history of metaphysics shows - there are two camps of thought about the physical world. Broadly it divides into the reductionism of atomism and the holism of a relational or systems approach.

The systems approach is triadic and says reality is a self organising hierarchical structure. So it deals with the potential and the actual by saying, yes, well both exist. It ain’t either/or. You have being and becoming. This makes sense as you also have necessity to complete the triad.

This triadic metaphysics reveals itself in many guises. Aristotle’s hylomorphism, Hegel’s dialectics, Peirce’s semiotics. So it gets confusing. But it offers a structure that can be used to see that modern physics is recapitulating the same holistic moves. The Planck triad of constants is a key sign of that.

So I am arguing from a particular metaphysical point of view. And I am saying reductionism is a monistic, concrete and object-oriented metaphysics in which the concept of potential doesn’t even make sense. Reductionism is comfortable with a world in which things either exist or fail to exist. Atoms and void. The nearest reductionism gets to a state of potential is allowing for some concrete ensemble of statistical possibilities.

This entified approach to existence carries over to talk about dimensionality. Dimensionality doesn’t develop. It simply exists, or fails to exist, in a concrete countable way. A point is a 0D object. A line is a 1D object. End of discussion. There is no “why” as to how it might be the case, given the limited ontological options that reductionism employs.

It you are fed up with hitting that brick wall, then that is where a systems metaphysics is worth a spin. It takes the bigger story seriously. It offers models that speak to the notion of pure potentiality - as that which can then birth self-organising reciprocal limits - in things like Anaximander’s Apeiron, Aristotle’s prime matter, Peirce’s logic of vagueness, the quantum foam of quantum gravity physics.

So I have urged stepping back and considering how one could even talk about continua except within the limits of a dichotomy - the dichotomy of the discrete~continuous.

You are talking about the fundamentality of the interval - a concrete or actual object that is a finite length that is thus both discrete and continuous at the same time. Well how does the finite interval get to be a combination of apparently contradictory properties. How could that state of affairs develop from something more fundamental? What is the deeper story on how finitude itself can arise?

So the choice is not between the zero-D point that makes no ontological sense and the truncated 1D interval that suddenly makes sense. You can claim to have no problem with an infinity of cuts and yet have a problem with an infinity of points.

I would say the 0D point and truncated interval are in the same class of question-begging objects. Both are atomised entities lacking a properly motivated existence.

If you want to move the argument forward, well at least talking about truncated intervals highlights the contradiction that a reductionist metaphysics embodies, but a systems metaphysics seeks to explain.

How do the discrete and the continuous combine in the one actualised object? How does such a state of affairs develop? And out of what?

I'm not sure how your post relates to my quote. — jgill

You said, "I have never used infinity as anything more than unboundedness."

I don't opine as what 'used' means there. I only pointed out that the calculus uses certain infinite sets, even if not explicitly. Just the real line alone is based on having the infinite set of real numbers.

It's just that conversations involving cardinalities beyond ℵ1 don't usually occur in classical or even much of modern analysis. — jgill

(1) I only said that infinite sets are used. I didn't say infinite sets with cardinality greater than aleph_1 are used.

(2) Just to be clear, we don't know what aleph is the cardinality of the set of real numbers.

(3) What is the difference you have in mind between classical and modern? Ordinary contemporary analysis is classical analysis.

/

As to the Wikipedia quote, of course I agree, and I mentioned the distinction between the notion of infinite size and the notion of points of infinity.

Clarification:

When you said "I have never used infinity as anything more than unboundedness", perhaps I misunderstood you. I thought you meant 'infinity' in the sense of infinite sets. That is, I thought you meant that you recognize that certain sets are infinite, but you don't use them.

But maybe you didn't mean that you don't use those sets. But that you do use them, but you don't use the extended real line with its points of infinity? As instead you simply deploy the fact that the reals are unbounded?

But maybe you didn't mean that you don't use those sets. But that you do use them, but you don't use the extended real line with its points of infinity? As instead you simply deploy the fact that the reals are unbounded? — TonesInDeepFreeze

The basic set theoretic structure of the reals underlies almost everything I have done, but I haven't used infinity as a "point" (nor the axiom of choice). Infinity is a limit in the language of calculus.

(3) What is the difference you have in mind between classical and modern? Ordinary contemporary analysis is classical analysis. — TonesInDeepFreeze

Classical means the tools of analysis like limits, differentiation and integration and all those entail. Nitty gritty. Actual specific results vs broad generalities. The more modern you get the more abstract the subject becomes with broad generalities and topological arguments. It's vague to an extent.

Hard & Soft Analysis

Let's imagine a line where cuts have been made to mark all rational points (I don't believe this is possible, but let's go with it for now). I believe you cannot mark any more points on this line. If you throw a dart in between the rational points then you will hit an indivisible line segment. That is as discrete as it gets, and even then the line is securely continuous. — keystone

The cuts are 0-dimensional so they are illusions of convenience. If you throw a dart at the line you will always hit the line, never the cut. The cuts have measure 0 after all. — keystone

The problem here is that the real number line is the mathematical object that was in question, surely? So as a construction, it hosts both the rational and the irrational numbers as the points of its line.

Now, like Peirce and Brouwer, you might want to make more ontic sense of this by employing the notion of intervals.

And so the claim becomes that reality has a fundamental length – the unit one interval. There is a primal atom of 1D-ness or continuity. There is an object or entity that begins everything by already being both extended and truncated. And in being both these things as a primal symmetry state – possessing a canonical oneness both as a contradiction and an equality – it can then become the fundamental length that then gets either endlessly truncated, or endlessly extended.

The equality of 1 can be broken by a move in either direction. And each move creates a ratio - a rational number - that speaks to the number of steps taken away from home base. If you can count upwards to a googol/1, then you can divide downwards to 1/googol. The reciprocal relation between extension and truncation is right there explicitly in the symmetry breaking of the unit 1 interval.

Now as reciprocal directions of arithmetic operations, each would seem to extend infinitely. Or at least, they are unbounded operations. The higher you can count with this system, the smaller you can divide its parts. But could you reach actual zero, or actual infinity? Well the obvious problem is that this would mean squeezing your origin point - the unit 1 - out of actual existence. The whole way of thinking would lose its anchoring conception of the truncated interval that set the whole game of approaching its dichotomous limits going.

So yes. If you just think of numbers as rationals – ratios that embody reciprocal actions on a fundamental length – then actual completed infinities and actual 0D points or cuts become anathema to the metaphysical intuition.

What then happens when you add the irrationals? Does that change anything?

The usual way of picturing it is that the number line becomes so infinitely crowded by markable points that it is effectively a continuum. Infinite extension and infinite truncation become the same thing. A new state of symmetry. A new unit 1 state. The continuum is that which is neither truncated nor extended. The concept of a finite length anchoring things is dissolved and becomes vague.

In hierarchy theory terms – Stan Salthe's "basic triadic process" – this is a familiar state of affairs. The small has become so small that it just fuses into a steady blur. The large has by the same token become so large that it has completely filled the field of view.

You wind up in a world where there is a global bound that arises because continuity has been extended so much that its truncated ends have crossed the event horizon (as cosmology would call it). And likewise, the local bound has become so shrunk in scale that it is a fused blur of parts (a wavefunction as quantum physics would call it). And then that leaves us, as the observing subject, surrounded by a bunch of medium sized dry goods – objects that embody both truncation and extension in terms of looking like composite wholes made of divided parts.

So we can make sense of the rationals as intervals on a continuum – the symmetry of the unit 1 being equally broken in both its complementary directions. And then when this asymmetry is maximised, you wind up in this hierarchical situation where you live in a world of truncated lengths, but then are semiotically closed in by a global bound of holographic continuity (synechism or constraint, in Peirce speak) and a local bound of quantum discreteness or fluctuation (tychism or spontaneity, in Peirce speak).

But that is connecting with the physics. A richer notion of mathematics that includes time and energy along with space. I'll get back to the issue of the irrationals.

Now for me, it seems clear enough that the familiar irrational constants – pi, phi, e, delta – are again unit 1 ratios, but ratios generated by growth processes. So e for example scales the compounding growth of a unit 1 square, not a unit 1 interval. That is why it is incommensurate. It is a unit 1 dropping in on the numberline from a higher dimensionality.

Surds have the same story. That then leaves the unbounded cloud of numbers with random decimal expansions. Some like pi, phi, e, delta have their generators in a higher dimension as I argue. But that isn't even a drop in the ocean of all the infinitesimals that seem to exist and so make the numberline infinitely dense with dimensionless points or cuts.

One view that appeals is that all these meaningless numbers with random decimal expansions mark nothing in particular. They are in fact the tychic spontaneity of Peircean semiotics – the inability of nature to suppress or constrain all its surprises. Down at the ground level of truncation, it is just fluctuation – the seething instability of the quantum vacuum, filled with its virtual particles and zero point energy (to use some much abused terms).

But maybe also, all these random fluctuations are actually ratios of some kind – visitors from another dimension like the growth constants. Maybe they all have a generator that makes each of them a unit 1 story in a bigger picture, one with infinite dimensionality. Or unbounded dimension.

So the infinitesimals becomes a blur of unit 1-ness that results from a numberline living in infinite relational dimensionality, just as Cantorian infinity establishes its own unit 1-ness by becoming the point where counting goes "supra-luminal" – crosses the event horizon in physical terms.

Something is certainly going on here with the irrationals. Some definitely have their higher dimensional generators like the unit square, the unit rectangle, the unit circle, the unit Pythagorean triangle. Crisp and necessary mathematical structure arises out of the swamp of algebraic symmetry breaking.

Do the rest of the irrationals have a similar story of geometric necessity behind them? Or are they just a blur of surprises and accidents. A blur of differences that don't make a differences and hence that which serves as the blank and continuous backdrop to the constants of nature which in turn have the most supreme importance.

I think Peirce and Brouwer argue towards a world where the number line starts off from the conception of truncated intervals – the infinity of truncation~extension operations that can act on the unit 1 length. That gives us the rationals that are completely at home in the world so defined.

Then you get the intrusions from higher mathematical dimensions – ratios or symmetry breakings from a larger universe of rational shapes. These pop on the line in ways that don't fit so exactly.

This then leads to a view of the number line that is a continuum of fluctuations. But now we are counting all digits with random decimal expansions as a something rather than a nothing.

It is information theory all over again where both meaning and nonsense are assigned the same bit-hood status. Signal and noise are in the eye of the beholder. What matters in the new counting system is there is the truncated interval – the ensemble of microstates. Information theory becomes about counting all actualised differences, not the differences that make a difference (even if that dichotomy can then be recovered with other relational measures like the notion of mutual information).

Anyway, the picture I have in mind is a continuum which is composed of rational intervals – truncated lengths that appear over all possible length scales. Eventually these lengths become either too large or too small and so exceed our pragmatic grasp. We live in a world where energy and time matter, along with (3D) space. And so it is simply a fact of living in that world which imposes a cosmic event horizon and a Planck scale cut-off. As an intuitionist would say, we haven't got the time or energy to pursue either the promise of unbounded interval expansion or unbounded interval truncation.

But into this rational continuum creeps some irrational numbers that really matter – the rare visitors from higher-D ratios.

And also then, we have the noisy background crackle that is the randomness of all the other unbounded decimal expansions we can imagine as truncation operations. If they have generators, then in the spirit of Kolmogorov complexity, there is no actually simpler algorithm available but to print out every entropic digit. They represent the limit of the generatable. Another way of saying they are merely the meaningless noise that can't be squeezed out of any system. The quantum uncertainty that one knows one finds at the Planck cut-off of any system made of actual material stuff.

The basic set theoretic structure of the reals underlies almost everything I have done, but I haven't used infinity as a "point" — jgill

Makes sense.

Classical means the tools of analysis like limits, differentiation and integration and all those entail. Nitty gritty. Actual specific results vs broad generalities. — jgill

That differs from how I find 'classical' is used. I find that 'classical' mathematics means all and only those results that can be formalized as theorems of ZFC with classical logic. And classical logic means the first order predicate calculus including the law of excluded middle.