Theists Equivocating the Empirical with the Transcendental

[Skepdick]

Falsificationism works really well in science.
It is not a philosophy that would be applicable to mathematics, though.

You would be surprised to learn that falsification (counter-examples) works just as well in Mathematics as they do in science.
All swans are white is falsified by a single black swan.

And the below is the single shortest paper ever published.

counter-example.jpeg (30.1 KiB) Viewed 755 times

While science is about the physical universe and tries to account for causality, mathematics is not about the physical universe and does not deal with the notion of causality. There simply is no "Theorem of Causality" in arithmetic theory or set theory.

That is news to me. I thought Mathematics was about entailment, but A entails B and A causes B can be formalised in exactly the same way: A ⊢ B.

And not to get too bogged down in any particular notation A ⊢ B can be trivially re-written as: enitails(A,B) -> Bool - a binary function which returns a Boolean.

Much like causality can be written as causes(A, B) -> Bool - a binary function which returns a Boolean.

(Mathematical) constructivism is philosophical concern in mathematics. It is not a treatise on general epistemology.

Oh yeah? What do you think model-construction is all about?

https://en.wikipedia.org/wiki/Construct ... f_science)

Formally, if we want a witness, we mean:

In mathematical logic, a witness is a specific value t to be substituted for variable x of an existential statement of the form ∃x φ(x) such that φ(t) is true.

That is all it is. It is not possible to transpose this mathematical notion verbatim to epistemology.

Oh really?

Let X be an unbound variable in the domain of humans.
Let φ be the English predicate has_six_toes(X) - for any particular human the predicate is either true; or false.

The Mathematical statement "∃x φ(x) such that φ(t) is true" translates into English thus: There exists a person (X) such that has_six_toes (X) is true.

I hereby substitute the variable X with this witness. I bind the variable X to the following particular human..

extra-toes-1.jpeg (13.08 KiB) Viewed 754 times

[godelian]

You would be surprised to learn that falsification (counter-examples) works just as well in Mathematics as they do in science.
All swans are white is falsified by a single black swan.

They work for conjectures but not for theorems.
Falsificationism potentially always works for science because there are no theorems in science.

That is news to me. I thought Mathematics was about entailment, but A entails B and A causes B can be formalised in exactly the same way: A ⊢ B.

Yes, the formalisms are quite compatible. However, the term causality is not in use in mathematics.

Concerning the term constructivism, it is indeed also in use in other knowledge fields.

In contemporary mathematics, the terms is mostly limited to demanding a witness (an example). Non-constructivist proofs still exist and are accepted. It is just that constructivist ones are preferred when they are uberhaupt possible.

[Veritas Aequitas]

What I am referring to is the Philosophy of Mathematics, i.e. upon reflection all modern mathematics are reducible to its roots in the empirical, e.g. numbers which are abstracted from the human empirical sense.

Modern mathematics is completely divorced from human empirical sense. It is symbol manipulation only, which does not need any empirical connection. In fact, it is preferable to reject such connection. If we can do pure reason, why would we reason with impurities? In modern mathematics, you must let go of physical reality. It is simply irrelevant:

In the philosophy of mathematics, formalism is the view that holds that statements of mathematics and logic can be considered to be statements about the consequences of the manipulation of strings (alphanumeric sequences of symbols, usually as equations) using established manipulation rules. A central idea of formalism "is that mathematics is not a body of propositions representing an abstract sector of reality, but is much more akin to a game, bringing with it no more commitment to an ontology of objects or properties than ludo or chess."[1] According to formalism, the truths expressed in logic and mathematics are not about numbers, sets, or triangles or any other coextensive subject matter — in fact, they aren't "about" anything at all.

Modern mathematics does not represent "an abstract sector of reality". I do not have a problem that they teach the kids aboriginal mathematics at school. Just like the pre-Aristotelian Egyptian mathematics, it is moderately useful, even though they could also just learn to use a calculator, instead of making endless procedural exercises in manual arithmetic. It simply does not teach them modern mathematics. It is not even a good beginning.

I believe your thinking is too shallow here.
You are implying what I am stating is;
a professor teaching modern mathematics will resort to making reference to his "fingers" when making reference to numbers.
This is very silly.

As I had reminded you, I am discussing the Philosophy of Mathematics where I insisted whatever the mathematics they are traceable to an 'empirical' root.

My use of 'empirical' was perhaps confusing.
Note, when I used the term 'empirical' [for convenience sake at your level] I do not confine it to the typical definition but rather to be more precise it is with reference to the 'world of senses' 'human conditions' and the a priori, not a posteriori [post experience].

Note Kant on the Philosophy of Mathematics;

Arithmetical Propositions are therefore always Synthetic.

The Concept of 12 is by no means already Thought in merely Thinking this union of 7 and 5; and I may analyse my Concept of such a possible sum as long as I please, still I shall never find the 12 in it.
We have to go outside these Concepts, and call in the aid of the Intuition which corresponds to one of them,
our five fingers, for instance,
or, as Segner 1 does in his Arithmetic, five points, adding to the Concept of 7, unit by unit, the five Given in Intuition.
For starting with the number 7,
and for the Concept of 5 calling in the aid of the fingers of my hand as Intuition,
I now add one by one to the number 7 the units which I previously took together to form the number 5,
and with the aid of that figure 2 [the hand] see the number 12 come into being.
That 5 should be added to 7, 3
I have indeed already thought in the Concept of a ‘sum=7 + 5,’
but not that this sum is equivalent to the number 12.
[CPR B15-16]

From the above, it implies there are no platonic '5' '7' or '12' existing independently by themselves to be discovered by humans.

Thus my point is whatever the mathematics, it has as least some link to the human conditions, i.e. a priori - the basis of the empirical.

Application to celestial mechanics

Ok. Point conceded. I am not familiar with celestial mechanics, and I didn't know that they have a use for the quintic in it. Subjects such as astronomy are about physical reality, and that is why I am not familiar with them, and why I do not like them particularly much. I am committed to pure abstraction. Therefore, I reject the physical universe as something that would be particularly interesting. That is why I pride myself on ignoring physical reality. I just don't like being too attached to it.

That would be delusion to deny reality-as-it-is which is physical [scientific] supported by philosophical reasonings.

The metre is currently defined as the length of the path travelled by light in a vacuum in
1 / 299 792 458 of a second.
Thus my point again, the fundamentals of mathematics [old or new] is reducible to the empirical.

That is physics. That is not mathematics. Physics is about the physical universe. Mathematics is not.

That is an example of how everything regardless is traceable to the human conditions.

Even if you can come up with illusory mathematics via pure reason or whatever, the fundamental is still empirical.

Again, you confuse the concept of "abstract" with "illusory".

The fundamentals of Peano Arithmetic Theory are a set of rules that consist of symbols. The fundamentals of Zermelo-Fraenckel Set Theory are a set of rules that consist of symbols. Where did you ever see such symbols in physical form in the physical universe?

Whatever the map, i.e. the mathematical model, its direct intention is to map the territory, i.e. the physical empirical world, else one is in la la land.

These things are Platonic abstractions. They exist but not in the physical universe. They exist in their own separate world.

A major question considered in mathematical Platonism is: Precisely where and how do the mathematical entities exist, and how do we know about them? Is there a world, completely separate from our physical one, that is occupied by the mathematical entities? How can we gain access to this separate world and discover truths about the entities? One proposed answer is the Ultimate Ensemble, a theory that postulates that all structures that exist mathematically also exist physically in their own universe.

Platonic ideas, form and universals are highly contentious and easily demonstrated to be unreal.

Kant had demonstrated Plato ended up in la la land; [mine]

Misled 1 by such a proof of the Power of Reason, the demand for the extension of Knowledge recognises no Limits.
The light dove, cleaving the air in her free flight, and feeling its resistance, might imagine that its flight would be still easier in empty Space.

It was thus that Plato left the World of the Senses, as setting too narrow Limits to 2 the Understanding, and ventured out beyond it on the wings of the Ideas, in the empty Space of the Pure Understanding.
He [Plato] did not observe that with all his efforts he made no advance meeting no resistance that might, as it were, serve as a support upon which he could take a stand, to which he could apply his powers, and so set his Understanding in motion.
CPR -[A5] [B9]
[Plato who flew into la la land & fantasy]

Thus the idea that mathematical entities exist by themselves out there is absurd.

What is most realistic is we can abstract and establish mathematical models from the empirical world.
And such abstracted model but eventually must make sense with the empirical world, e.g. the 'quintic' example above.
However for a higher philosophical deliberations, all mathematical models are traceable to its roots in entanglement with the human conditions i.e. the a priori.

Where you insist mathematical entities [platonic] exist as real by themselves independent of human conditions, that is delusional.

Btw, which "shoulders of giants" [philosophers] are you standing on to support your views above?

[Skepdick]

They work for conjectures but not for theorems.
Falsificationism potentially always works for science because there are no theorems in science.

Errr, what?

https://en.wikipedia.org/wiki/Noether%27s_theorem
https://en.wikipedia.org/wiki/No-cloning_theorem
https://en.wikipedia.org/wiki/Bell%27s_theorem
https://en.wikipedia.org/wiki/Category: ... _mechanics
https://en.wikipedia.org/wiki/Category: ... relativity

Yes, the formalisms are quite compatible. However, the term causality is not in use in mathematics.

Well, if the term triggers you lets call it entailment and not causality.

In contemporary mathematics, the terms is mostly limited to demanding a witness (an example).

Well, if the term triggers you lets call it a witness, not evidence.

Non-constructivist proofs still exist and are accepted. It is just that constructivist ones are preferred when they are uberhaupt possible.

Yes but proofs by contradiction are not accepted. Or rather - that's a half-truth.

You can assume P, derrive a contradiction and conclude not-P - that's now you constructively prove a negative.
What you cannot do is assume not-P, derrive a contradiction and conclude P - which is how existence is currently proven in classical mathematics.

Thre parallels to empiricism are obvious. Constructivism requires positive evidence, classical mathematics settles for absence of negative evidence.

[godelian]

The Concept of 12 is by no means already Thought in merely Thinking this union of 7 and 5; and I may analyse my Concept of such a possible sum as long as I please, still I shall never find the 12 in it.
We have to go outside these Concepts, and call in the aid of the Intuition which corresponds to one of them,
our five fingers, for instance,

From the above, it implies there are no platonic '5' '7' or '12' existing independently by themselves to be discovered by humans.

If you want to understand the nature of the standard universe of the natural numbers, read up on Peano Arithmetic Theory (PA) and the Lowenheim-Skolem theorem which will give you an introduction to modern model theory and which clarifies that there exist nonstandard universes that interpret PA, and therefore, that PA generates a multiverse:

In mathematical logic, a non-standard model of arithmetic is a model of (first-order) Peano arithmetic that contains non-standard numbers. The term standard model of arithmetic refers to the standard natural numbers 0, 1, 2, …. The elements of any model of Peano arithmetic are linearly ordered and possess an initial segment isomorphic to the standard natural numbers. A non-standard model is one that has additional elements outside this initial segment. The construction of such models is due to Thoralf Skolem (1934).

There are several methods that can be used to prove the existence of non-standard models of arithmetic.

- From the compactness theorem.
- From the incompleteness theorems.
- From an ultraproduct.

Therefore, the idea that the only interpretation of arithmetic theory would be the natural numbers, is utterly simplistic and even ignorant. Furthermore, it is not possible to discover the existence of non-standard numbers by counting your fingers. Again, there are serious limits to aboriginal empiricism. Therefore, we cannot use that approach in modern mathematics. Aboriginal empiricism must be utterly rejected.

Furthermore, there is no way that you can understand that there exist logic sentences that are true in the natural numbers but not provable from PA (Gödel's incompleteness theorem), if you stick to your 18th century view on arithmetic. Kant's simplistic analysis of the natural numbers has been completely superseded in the two centuries that followed the publication of CPR.

A major question considered in mathematical Platonism is: Precisely where and how do the mathematical entities exist, and how do we know about them? Is there a world, completely separate from our physical one, that is occupied by the mathematical entities? How can we gain access to this separate world and discover truths about the entities? One proposed answer is the Ultimate Ensemble, a theory that postulates that all structures that exist mathematically also exist physically in their own universe.

Platonic ideas, form and universals are highly contentious and easily demonstrated to be unreal.
Kant had demonstrated Plato ended up in la la land; [mine]
Thus the idea that mathematical entities exist by themselves out there is absurd.
Where you insist mathematical entities [platonic] exist as real by themselves independent of human conditions, that is delusional.
Btw, which "shoulders of giants" [philosophers] are you standing on to support your views above?

You seem to dislike mathematical Platonism. However, that does not matter, because that will not change anything to the fact that it is the dominant ontology for mathematics amongst mathematicians.

Platonism in the Philosophy of Mathematics

Platonism about mathematics (or mathematical platonism) is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices. Just as electrons and planets exist independently of us, so do numbers and sets. And just as statements about electrons and planets are made true or false by the objects with which they are concerned and these objects’ perfectly objective properties, so are statements about numbers and sets. Mathematical truths are therefore discovered, not invented.

The most important argument for the existence of abstract mathematical objects derives from Gottlob Frege and goes as follows (Frege 1953). The language of mathematics purports to refer to and quantify over abstract mathematical objects. And a great number of mathematical theorems are true. But a sentence cannot be true unless its sub-expressions succeed in doing what they purport to do. So there exist abstract mathematical objects that these expressions refer to and quantify over.

1. What is Mathematical Platonism?

Mathematical platonism can be defined as the conjunction of the following three theses:

Existence.
There are mathematical objects.

Abstractness.
Mathematical objects are abstract.

Independence.
Mathematical objects are independent of intelligent agents and their language, thought, and practices.

1.2 The philosophical significance of mathematical platonism

Mathematical platonism has considerable philosophical significance. If the view is true, it will put great pressure on the physicalist idea that reality is exhausted by the physical. For platonism entails that reality extends far beyond the physical world and includes objects which aren’t part of the causal and spatiotemporal order studied by the physical sciences.[1] Mathematical platonism, if true, will also put great pressure on many naturalistic theories of knowledge. For there is little doubt that we possess mathematical knowledge. The truth of mathematical platonism would therefore establish that we have knowledge of abstract (and thus causally inefficacious) objects. This would be an important discovery, which many naturalistic theories of knowledge would struggle to accommodate.

2.4 The notion of ontological commitment

Versions of the Fregean argument are sometimes stated in terms of the notion of ontological commitment. Assume we operate with the standard Quinean criterion of ontological commitment:

Quine’s Criterion.
A first-order sentence (or collection of such sentences) is ontologically committed to such objects as must be assumed to be in the range of the variables for the sentence (or collection of sentences) to be true.

Then it follows from Classical Semantics that many sentences of mathematics are ontologically committed to mathematical objects.

So, according to your own remarks, you know absolutely nothing about mathematical Platonism, while you use the fact that you are completely ignorant of mathematical Platonism as the justification for why it would be wrong. So, no, your views do not not justify that mathematical Platonism would be wrong. Your ignorance on the matter only justifies the claim that you know nothing about it.

[Iwannaplato]

Therefore, the idea that the only interpretation of arithmetic theory would be the natural numbers, is utterly simplistic and even ignorant. Furthermore, it is not possible to discover the existence of non-standard numbers by counting your fingers. Again, there are serious limits to aboriginal empiricism. Therefore, we cannot use that approach in modern mathematics. Aboriginal empiricism must be utterly rejected.

Pardon an intrusion in your exchange with VA, but it seems to me some of the argument might be beside the point. Even if math in humans began with empirical math, counting things, there is math that is not empirical. I know you have said this. He will keep trying to make it seem like there is nothing transcendant by saying that it all stems back to counting, say. But that is irrelevant to any mathematician who is remotely Platonic. Humans have also found math, but noticing relations in reality: physics. It seems like the universe runs on equations, but they are not present like chairs or stones, these mathematical 'things'. So, many mathematicians are platonists, believing that math runs things, from behind, in some ideal form or one might use other language to mean, in the end something not manifest, transcendant. Further there is math that is not connected to the empirical, so there are things that are not connected. It doesn't matter where it came from in
humans
processes of
finding it.

Ultimately he wants to say that the transcendant is impossible. So, it doesn't matter where math disconnected from the empirical came from if it has transcendent qualities, as would some parts of math and then also in the equations relationship with matter. They are not matter, but affect matter.

I am guessing you know all this, but in my reading of your exchange with VA, it has seemed to me like he is sort of distracting things through is 'but it really comes from.....' and other tactics. I don't mean he is disingenous, but still I think there is something repeatedly fallacious about his arguments.

I cannot find a good source, but many mathematicians think it is likely that there are more [edit] Platonists than Nominalists in their group.

[Skepdick]

You seem to dislike mathematical Platonism. However, that does not matter, because that will not change anything to the fact that it is the dominant ontology for mathematics amongst mathematicians.

Saying you are a Platonist doesn't make you a platonist any more than saying you are a cow makes you a cow.

We can talk about Platonism in the abstract. We can say that we are Platonists while we play the language game, but in practice there are no Platonists - the set of Platonists is uninhabited because it is an undecidable problem whether any particular human is a Platonist.

So, according to your own remarks, you know absolutely nothing about mathematical Platonism, while you use the fact that you are completely ignorant of mathematical Platonism as the justification for why it would be wrong.

Mathematically speaking, what does it mean for any philosophical view to be "wrong"?

I have no idea if it's "right" or "wrong" - it's just incoherent.

How do you account for the interaction between the Platonist and the Platonic objects?
How does an epistemologist discover ontological Platonic objects and attain knowledge about them if they are not empirical?

You are not wrong about Platonism. You are just wrong about your identity as a Platonist. You would understand that if you understood identity types.

[bobmax]

And I believe in falsification. Platonism (if it claims to be separate) cannot be correct. Because it fails to account for causality.

"Separate" universes that are causally interacting are not separate.

I think Plato was one of humanity's greatest logicians.
However, also for this reason, he was also "the parricide" of Parmenides.

Plato represents a turning point in the world view.
When the rational interpretation of reality has taken over.

The world of Platonic ideas, although invented to save the transcendent, is the tombstone of the wonderful pre-Socratic philosophy.

We have therefore come to the point of believing, for example, in the existence of numbers regardless of the mind that thinks them!

A serious misunderstanding that has led to a fundamental impetus for the growth of nihilism.

Cantor's much-acclaimed diagonal argument is clear evidence of this.
Infinity does not exist!

Instead Cantor and most mathematicians now treat infinity as if it were a thing.
Nihilism...

[Iwannaplato]

Show me your two decision procedures please.

One decision procedure for roundness; and one for perfect roundness. If you can't do that I am not sure what mathematical meaning you ascribe to the English word "perfect".

A circle is made up of an infinite amount of points. I don't think modern physics supports a thing being made that has an infinite amount of points on it nor that it could be accurately equidistant on all points (or stable) in a quantized universe, given how quanta act.

Also, if you are assuming zero-width points you are certainly not working in any system with infinitesimals (numbers which are potentially zero, but not zero). A rather arbitrary choice of axioms...

I am not assuming anything. I am going by the standard definition of a point in geometry.

A point in geometry is a location. It has no size i.e. no width, no length and no depth. A point is shown by a dot. A line is defined as a line of points that extends infinitely in two directions

And yeah, the axioms of geometry are too a great extent arbritrary. And yeah, they aren't like empirical objects that we can measure. That's why they can't be made. And I don't see what that synthetics mathematics had to do with anything since your argument is basically we can make something good enough that, maybe, our measuring devices won't notice imperfections. Though they would, since right off they could rule out infinite points, not just since they go against theory, but in an extremely fine circle like you are proposing, they could actually look at individual particles and count them.

But we can rule out their existing. Unless there is a radical change in physics, which has happened before and could happen again. But right now a geomtric circle can't be made or found.

Here's a blog post that might help you along... http://math.andrej.com/2008/08/13/intui ... r-physics/

Ah, now I remember how you are 'help you along'. And we were so civil without your little dominance, condescension games, when you're being half subtle. Like, what is that crap. Does it make you feel more right when you play these games? Does it tend to make people less careful posters in response to you, and this distracts from the actual discussion? and, ya, sure, this was mild, by your standard levels of crap...

Whatever it is, it's not necessary. I'll ignore you until I forget why I was ignoring you again or don't forget and it keeps.

[godelian]

Even if math in humans began with empirical math, counting things, there is math that is not empirical.

Agreed. In fact, we can call modern mathematics also "Aristotelian mathematics", since it is Aristotle who started insisting on foundationalism in mathematics:

2. The Structure of a Mathematical Science: First Principles

Aristotle's discussions on the best format for a deductive science in the Posterior Analytics reflect the practice of contemporary mathematics as taught and practiced in Plato's Academy, discussions there about the nature of mathematical sciences, and Aristotle's own discoveries in logic. Aristotle has two separate concerns. One evolves from his argument that there must be first, unprovable principles for any science, in order to avoid both circularity and infinite regresses. The other evolves from his view that demonstrations must be explanatory. (See subsections A, B, and C of §6, Demonstrations and Demonstrative Sciences, of the entry Aristotle's logic.) Aristotle distinguishes (Posterior Analytics i.2) Two sorts of starting points for demonstration, axioms and posits.

This is a first blow to the nominalist-empiricist view on mathematics, as modern mathematics does indeed implement Aristotle's foundationalism completely and throughout. Modern mathematics reasons from unproven First Principles, i.e. axioms, that inevitably reflect a notion of Blind Faith in order to produce non-empirical conclusions. Modern mathematics is therefore Pure Reason.

To that effect, mathematics had to be completely restructured onto a foundation of unprovable principles and abandon empiricism.

Further there is math that is not connected to the empirical, so there are things that are not connected. It doesn't matter where it came from in
humans processes of finding it.

Agreed. Pursuing the foundationalist approach to modern mathematics has led to the discovery of model theory, which strongly suggest the existence of an abstract, Platonic universe generated by its axiomatic theories.

Ultimately he wants to say that the transcendant is impossible.

Yes, indeed. Some people are so committed to their attacks on religion that they completely reject the existence of non-physical universes, even mathematical ones.

I cannot find a good source, but many mathematicians think it is likely that there are more [edit] Platonists than Nominalists in their group.

Yes, Platonism is generally considered the dominant ontology in mathematics:

Mathematical platonism enjoys widespread support and is frequently considered the default metaphysical position with respect to mathematics.

This is unsurprising given its extremely natural interpretation of mathematical practice. In particular, mathematical platonism takes at face-value such well known truths as that “there exist” an infinite number of prime numbers, and it provides straightforward explanations of mathematical objectivity and of the differences between mathematical and spatio-temporal entities. Thus arguments for mathematical platonism typically assert that in order for mathematical theories to be true their logical structure must refer to some mathematical entities, that many mathematical theories are indeed objectively true, and that mathematical entities are not constituents of the spatio-temporal realm.

Still, I do not reject formalism, structuralism, or even logicism. As I see it, these alternative ontologies emphasize a different but equally compelling aspect of mathematics.

It is a bit like the fact that the following two sentences are not contradictory:

- All mathematical objects are in fact numbers.
- All mathematical objects are in fact sets.

At first glance, it sounds like both views exclude each other, but in fact, they do not.

In their 2007 publication, "On Interpretations of Arithmetic and Set Theory", Richard Kaye and Tin Lok Wong prove that arithmetic theory (PA) and set theory (ZF-inf) are bi-interpretable. This means that for every true logic sentence that is written in the language of natural numbers (PA) there exists a true logic sentence written in the language of sets (ZF-inf) that has equivalent semantics. Hence, all mathematical objects are simultaneously numbers and sets.

There is currently no academic paper that explores a notion similar to "bi-interpretability of axiomatizations" but then applied to ontologies (such as Platonism, formalism, structuralism, and logicism) but I sense that there must somehow exist such undiscovered notion.

[godelian]

Infinity does not exist!

I don't see why infinity would be a problematic notion.

If you look at the set { 2, 5, 8 }, you can see that it has 3 elements. In mathematical lingo, this set has a cardinality of 3.
Imagine that you put all the natural numbers inside a set: { 0, 1, 2, 3, 4, ... } (assuming that zero is a natural number).
How many elements does that set have? Well, very large, because the progression never stops.
Now, natural numbers are not physical objects. Therefore, there is no compelling reason to assume that it would take time, effort, or energy to traverse them from beginning till end. Let's assume that we can fully traverse them. Now let's name the cardinality of this set: countable infinity ("aleph-0").

Note that all the objects mentioned above are abstractions. The natural numbers are. Their set is. This set's cardinality is.
These objects are non-physical. None of these objects exist in the physical universe.
Countable infinity is therefore merely a similar abstract object, not much different from the other ones involved in this view.
Countable infinity is therefore not more abstract or less abstract than for example the natural number 10.

Cantor's much-acclaimed diagonal argument is clear evidence of this.

Cantor first produces a list of all real numbers. Then he shows that from this list you can always create a new number that is not in the list.
Hence, it is not possible to create a list of real numbers. Hence, the real numbers are uncountable.
Hence, the cardinality for the set of real numbers is necessarily larger than the cardinality for the set of natural numbers.
Cantor's diagonal argument is amazing because it is so simple.

Instead Cantor and most mathematicians now treat infinity as if it were a thing.

In an abstract, Platonic world filled with abstract objects such as numbers and sets, countable infinity does not even particularly stand out.
None of these things exist in the physical universe. If the number 5 does not exist in the physical universe, why should countable infinity exist in it?

People who are attached to an empirical concept of mathematics such as finger or toe counting, pretty much always have extreme problems with modern mathematics and its pure reason. They just don't get it! This is not a problem with modern mathematics. This is a problem with these people.

[bobmax]

Now, natural numbers are not physical objects. Therefore, there is no compelling reason to assume that it would take time, effort, or energy to traverse them from beginning till end. Let's assume that we can fully traverse them. Now let's name the cardinality of this set: countable infinity ("aleph-0").

It is this assumption that is wrong.
Because it is always possible to add another number.

Regardless of whether abstract or not, the set of natural numbers can never be infinite.
It is in the very nature of the infinite not to be there.

Infinity is an open idea, it cannot be taken and used as a thing.

It doesn't matter if abstract or not, infinity cannot be elaborated precisely because of its intrinsic impregnability.

Infinity cannot even be really thought of.
One always thinks only of the finite.

And it is always the finite which, by denying it, generates the infinite.
But only as a negation of the finite.
Not because you can really think of infinity...

Infinity is just pure negation.
And every negation draws all its raison d'etre in what it denies.

[Immanuel Can]

I don't see why infinity would be a problematic notion.

It's not...as a "notion" or concept, that is.

One can conceptualize all sorts of things, without any problem.

But the question is, can one have an actual infinite.

[Skepdick]

A circle is made up of an infinite amount of points.

For somebody who gets super-annoyed by the "domination game" you sure love to dominate discourse by forcing your (arbitrary) definitions onto the field and pretending that your definition is "standard", "normal"; or "widely accepted". I guess you don't like any competition in the game of "I reject your definitions and substitute my own".

The Mathematical universe I live in is nothing like the Platonic one - in my universe there are no ideal forms. Mathematics is absolutely relative and there are infinitely many Mathematical perspectives and definitions. In my perspective the phrase "the continuum" is a misnomer because THE implies there's only one continuum - there are as many continuums as one can define. Mathematical viewpoints across universes/perspectives are irreconcilable and there is no One True Universe. Not even mine. Now onto correcting your error...

Your conception of a circle is made up of infinite amount of points. Of course you forgot to (or chose not to) tell us which infinity you are even refering to. There are so many of them. So if I were to ask you about the cardinality of the set of points on your circle... what woud it be?

As far as I am concerned a circle is any object which satisfies the definition of a circle. Whose definition? MY definition. Which definition? Whichever definition I am interested in at any given moment for whatever purpose! Any similarity or intersection with your definition; or with the "standard" definition is purely coincidental - I don't give a shit about other people's definitions unless they are useful to me.

At this instant I happen to be interested in the definition x^2 + y^2 = 1; and I happen to be interested in the domain of x and y being the integers [-1,1]. So this particular circle is made up of just four points equidistant from 0,0.

circle.png (12.49 KiB) Viewed 630 times

And of course you will object to this being a circle and I will pay no attention to your objection.

I don't think modern physics supports a thing being made that has an infinite amount of points on it nor that it could be accurately equidistant on all points (or stable) in a quantized universe, given how quanta act.

Never mind physics. I don't think modern mathematics supports infinities. It pretends it does, but you can't DO an infinite number of operations.
We talk about infinities. We can finitely represent infinities when a set is compact. We can even search infinite sets in finite time.

But the fact remains. That circle you are speaking of doesn't exist, but I am willing to be proven wrong - go ahead and construct it.

I am not assuming anything.

That's a lie.

I am going by the standard definition of a point in geometry.

You are assuming that the geometric definition of a point is the "standard definition" of a point. Why?

Why not the topological definition? A topological space whose underlying set is the singleton,
Why not the category-theoretic definition? A point is a category with a single object.

There are so many definitions of points. You sure seem to be biase to some particular definition for some particular reason.

A point in geometry is a location.

Sure. Lets go with that. And location is an address in the context of a particular coordinate system.
And if your coordinate system is addressable then you have yourself a memory model (yes, the computational kind!) with each address/location corresponding to an address in memory.

So what sort of coordinate system do you have in mind when you are talking about points? As soon as you tell us we could model it computationally.

It has no size i.e. no width, no length and no depth.

That's one particular kind of a point. You can extend that point with whatever properties/attributes you like.

You want a point with size? Add size to it.
You want a point with length? Give it length.
You want a point with depth? Give it depth.

A point is shown by a dot.

A point. A circle with radius 0. What's the difference?

Here is a dot in 2 dimensions: x^2 + y^2 = 0.
Or if you want to move the dot around.... (x-2)^2 + (y+2)^2 = 0.
Or if you want to do this in three dimensions.... (x-2)^2 + (y+2)^2 + (z-8)^2= 0.

Or four (but at this point your geometric intuition becomes useless!) ... (x-2)^2 + (y+2)^2 + (z-8)^2 + w^2= 0.

The point I am making here is that an n-sphere with radius zero is a dot in ALL dimensions.

A line is defined as a line of points that extends infinitely in two directions

Again. Such a specific definition of a line from such a particular field of Mathematics. There are sooo many other definitions.

It's weird that you think this definition is special.

And yeah, the axioms of geometry are too a great extent arbritrary.

All axioms are arbitrary. Which is why there's no such thing as an Ideal Platonic circle or a line. There are infinitely many conceptions (definitions) of circles. That's merely the perspective of Euclidian geometry. But why have you chosen that particular geometry? There are soooo many other geometries to choose from!

And yeah, they aren't like empirical objects that we can measure.

The moment you construct a coordinate system in which all points are addressable you have yourself a computational model. It becomes empirical.
Because a location (address) trivially corresponds to a memory address in your computer's memory.

That's why they can't be made.

Sure they can. Just explain your definition to a computer and extract a proof-object.

And I don't see what that synthetics mathematics had to do with anything since your argument is basically we can make something good enough that, maybe, our measuring devices won't notice imperfections.

If you can formalize your definition well enough to explain it to a computer then you can reify your concept into a tangible thing.
The entity which you have defined exists as physical system. In particular - it exists in the memory banks of a computer. That "circle" or a "line" has a physical configuration and everything.

You can interact with it. You can manipulate it. You can experience it. You can compute with it. It's independent of your mind and everything.

Here's at least one software project with has those aspirations: https://geocoq.github.io/GeoCoq/

Though they would, since right off they could rule out infinite points, not just since they go against theory, but in an extremely fine circle like you are proposing, they could actually look at individual particles and count them.

But we can rule out their existing. Unless there is a radical change in physics, which has happened before and could happen again. But right now a geomtric circle can't be made or found.

So like. At this point you need to be called out for equivocating "existence" - you are riding the dualist fence.

If geometric circles don't exist then what is this "geometric circle" you are talking about.

What are you refering to if we take any and all definitions off the table? Nothing. That's what you are refering to. Definitions manifest circles into existence.

And we were so civil without your little dominance, condescension games, when you're being half subtle.

I am about as civil, dominating and condescending as you are when you keep pretending that there are such things as "standard definitions" for points, lines and circles.

The difference being you are completely oblivious to your attempt to dominate the discourse by appealing to a fictitious status quo.

Like, what is that crap. Does it make you feel more right when you play these games?

You are in a perfect position to answer that question all by yourself. I'd love to hear the answer.

Does it tend to make people less careful posters in response to you, and this distracts from the actual discussion? and, ya, sure, this was mild, by your standard levels of crap...

The discussion and the point has always been that philosophy is bullshit.

The idiot-philosophers spend far too much time studying definitions and far too little time studying the process of defining.

WHY do we define?
What are the limits of definition?

If you want to talk mathematics spare me the definitions and focus on definability.
If you want talk circles - spare me your definition of a circle and tell me what manipulations/operations you want to perform on the object you have in mind. Tell me which properties you are necessary for your purpose; and which properties are sufficient and which properties are irrelevant and out of scope.

Then we'll get to the meat ot the discussion and construct a model of a circle which satisfies your needs..

Whatever it is, it's not necessary. I'll ignore you until I forget why I was ignoring you again or don't forget and it keeps.

Ah well. Good riddance?

[Skepdick]

If you want to understand the nature of the standard universe of the natural numbers, read up on Peano Arithmetic Theory (PA) and the Lowenheim-Skolem theorem which will give you an introduction to modern model theory and which clarifies that there exist nonstandard universes that interpret PA, and therefore, that PA generates a multiverse:

It has always confused me (and in fact continues to confuse me) how anyone can accept the implications of the Lowenheim-Skolem theorem AND be a Platonist. At the same time.

Which Universe in the Multiverse of Mathematics is the Platonic Universe?