...and bad mathematics.

Common practice is pretty rigorous once you accept the axioms.

Well addition subtraction and the rest mean something for mathematicians, but if everything that is real is just infinite sets a lot of mathematics would become frivolous or unnecessary. My current position, I admit, does deny that numbers are real, but I think I have a point in that there could be a certain mathematics only of the infinite. I add and subtract for practical purposes but there doesn't feel like anything Platonic or ontological about it for me

Do you think mathematics of the infinite can be done without finite numbers?

I think of units of an infinite set as completely relational to the whole.

You're more educated in this than I

. . . but if everything that is real is just infinite sets . . . — Gregory

Whoa! That is quite an assumption. :scream:

Do you think mathematics of the infinite can be done without finite numbers? — Gregory

I don't mess with infinities. Others here do.

I read a lot of German idealism and I strongly feel they were arguing against the possibility of finite existence. Mysticism usually subsumes everything into infinities

Note: German thinkers since Kant (the Continentalists) generally preferred Parmenides, Heraclitus, and Spinoza over Pythagoras and Descartes

Hmmm. Live and learn. :chin:

Mathematics is rigorous by effectivized formal languages, recursive axiom sets, and recursive inference rules, and explicit statements of algorithms for checking for well-formedness, axiomhood, and proof.

Mathematics has both finite and infinite sets.

Your notion of a mathematics with only infinite sets is purely fanciful without at least an outline of the primitives and axiomatic notions.

Well this is a philosophy forum, not a math forum, so we need to try to stick to the basics of the philosophy of mathematics. Stick with me and read the following:

"It is thus not surprising that Hegel's book began with a devastating, even if very ironical, critique of Jacobi's position against Kantianism (and all the forms of post-Kantianism), namely that we are in possession of a kind of 'sense-certainty' about individual objects in the world that could not be undermined by anything else and which showed that there was an element of 'certainty' about our experience of the world that philosophy was powerless to undermine. Hegel called this a thesis about 'consciousness.' If we begin with our consciousness of singular objects and present to our senses (an awareness of 'things' that is supposedly prior to fully fledged judgments), and hold that what makes those awarenesses true are in fact the singular objects themselves, then we take objects to be the 'truth-makers' of our judgments about them; however in taking these objects to be the truth-makers of our awareness of them, we find that our grasp on them simply dissolves and the impetus for such a dissolution lies in the way we are taking them to play a role in consciousness. The result, Hegel argued, is that in the process of working out these tensions, we discover that it could not be the singular objects of sense-certainty that had been playing the normative role in making those judgments of sense-certainty true, but the objects of a more developed, more mediated perceptual experience had to have been playing that role all along... The dialectic inherent in Jacobi's sense-certainty thus turns on our being required to see the truth-making of even simpler judgments about the existence of singular things of experience as consisting of more complex unities of individual things-possessing-general-properties of which we are perceptually and not directly aware. That is, we can legitimate judgments about sigfular objects only be referring them to our awareness of them as sigfular objects possessing general properties, which in turn requires us to legitimate them in terms of our take on the world in which they appear as perceptual objects... We must acknowledge, as Kant put it, that it must be possible for an 'I think; to accompany all our consciousness of things." Terry Pinkard

So I am not trashing mathematics but, instead, probing it's assumptions. The view of many philosophers is that infinities alone exist and I'm wondering what happens to the rest of mathematics when this is accepted by someone. Infinities have cardinality and density, the former perhaps being bound to the latter and perhaps geometry as well. Hegel in particular thought mathematics eventually turns into theories of the infinite and pure logic and I thought it would be interesting to see other peoples' take on these questions

Sure, in a purely philosophical context, you can come up with all kinds of stuff.

But then mathematics is not one hundred percent truth. Even nowadays infinites things haven't all been worked out fully. Imagine a ruler going to infinity out into the horizon and into space forever. Is the ruler longer as a whole or equal to the odd sections? This illustrates that the relationship between cardinality and density can be tricky but then it sublates into a higher view that nothing is ultimately "longer" in a non-dualist ontology.

Sometimes novel ideas are helpful. I tried to start a discussion once about what it would mean if all math was wrong and the *opposite* of every equation and theorem was true. People didn't like that idea but I tried this thread out anyway. I recommend the essay "Holism and Idealism in Hegel" by Robert Brandom

then mathematics is not one hundred percent truth — Gregory

What do refer to when you say 'mathematics'?

What do refer to when you say 'mathematics'? — TonesInDeepFreeze

The theory of numbers. 1, then double one, ect. But a non-dualist or infinitist mathematics would no longer deal with number and I admit I don't have many details worked out, but I think it's an interesting idea in that it would think solely in terms of concentric infinities instead of finite units

The natural numbers?

They're not a doubling sequence. They're a successorship sequence.

And what theory of the natural numbers? There are many.

It's just about numbers in general vs sets which have infinite density

I tried to start a discussion once about what it would mean if all math was wrong and the *opposite* of every equation and theorem was true. — Gregory

That sounds exciting! Sorry I missed it. :lol:

It's just about numbers in general vs sets which have infinite density — Gregory

So anything anyone says about numbers in general is mathematics? And if your philosophizing about that mathematics is purely philosophical then that mathematics is not all true?

And what do you mean by 'infinite density'? Is that a mathematical notion of yours or purely philosophical?

Sometimes there is disenchantment with nature in math and science. But nature is truly enchanted. There is no such thing as a disconnected observer and the opposition within the dualism of "how things are" and "how I perceive them" can be overtaken by a holism in how we take things,

So anything anyone says about numbers in general is mathematics? And if your philosophizing about that mathematics is purely philosophical then that mathematics is not all true?

And what do you mean by 'infinite density'? Is that a mathematical notion of yours or purely philosophical? — TonesInDeepFreeze

From what I understand the terms "ordinal" and "density" are very much related. Wikipedia has articles on these terms but I don't know if you want to read into the math on this or not. HOWEVER, the philosophy of mathematics is a real field of study. I have books on it, and that is what this thread was about

Mathematics is rigorous by effectivized formal languages, recursive axiom sets, and recursive inference rules, and explicit statements of algorithms for checking for well-formedness, axiomhood, and proof. — TonesInDeepFreeze

Logically proving mathematics from the ground up was the incompleted project known as Logicism

I know what ordinals and density are. I just want to know what you mean by 'infinite density' in mathematics.

And I know that the philosophy of mathematics is a field of study.

Logicism is often thought to have failed because it was not found how to derive mathematics without non-logical axioms.

That does not refute what I said about the rigor of mathematics.

Imagine the ruler going into space infinitely again. The density of the whole is greater than the odd parts of the ruler but the cardinality is the same. Take a marble and the Earth now. They have equal number of parts in that they have the same uncountable infinity of subdivisions. But which is larger? My point is that math should rise our thoughts to higher stages by these thoughts instead of getting stuck in an infinite process of proof and review, especially considering Gödel's theorems. Math is not done apart from Nature. What do we mean when one thing is opposed to another?

I find no definite sense from your use of all that undefined terminology and assumptions. And I surmise that continuing to ask you will lead to only more.

But I am intrigued what you have in mind regarding Godel's theorems.

Ok. Mathematics is usually practiced as a Platonic type of religious practice. What did Plato espouse? Well he though there was a partition or doubling in reality with the shadow of the world on one side and intellectual reality on the other. These are one and yet two (very unmathematical in that sense) where one is the other of the other. I just see so much separation from the world with the union of math and science that I had to find Naturphilosophie to make sense of it. It's been widely reported that Neil de Grasse Tyson has caused a lot of depression in people (lol). One alternative is to say teleology is given by the divine to the world but another perspective is that the world is teleology and we are Nature. Anyway thanks for you insightful posts

Mathematics is usually practiced as a Platonic type of religious practice. — Gregory

No, it's not.

If they don't then they approach it from a dead universe perspective and will move in infinite circles (circles which, by the way, are better understood as infinites within Nature)

Would you give an example of a work in mathematics that is from a dead universe perspective and moves in infinite circles?