godelian wrote: ↑Tue Jun 21, 2022 10:05 am
Veritas Aequitas wrote: ↑Tue Jun 21, 2022 7:08 am
Kant in CPR wrote:
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:
Wikipedia on "nonstandard models of arithmetic" wrote:
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 Kant's simplistic analysis of the natural numbers has been completely superseded in the two centuries that followed the publication of CPR.
Veritas Aequitas wrote: ↑Tue Jun 21, 2022 7:08 am
Wikipedia on "mathematical Platonism" wrote:
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.
plato.stanford.edu on "Mathematical Platonism" wrote:
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.
Your mind and views are on a one-track-path to nowhere and la la land.
Note we are doing 'philosophy' here not Mathematics per se.
So what is of concern is the
Philosophy of Mathematics.
My basic principle re philosophy is,
All Philosophies are Reduced to Realism vs Anti-Realism [Idealism]
viewtopic.php?f=5&t=28643
Whilst you are banking on Platonic Mathematics, it is merely a type of
Mathematical Realism.
- Mathematical realism, like realism-in-general, holds that mathematical entities exist independently of the human mind. Thus, humans do not invent mathematics, but rather discover it, and any other intelligent beings in the universe would presumably do the same.
https://en.wikipedia.org/wiki/Philosoph ... al_realism
OTOH, my view of mathematics is that of
Mathematical Anti-Realism, e.g.
- In mathematics, intuitionism is a program of methodological reform whose motto is that "there are no non-experienced mathematical truths" (L. E. J. Brouwer).
From this springboard, intuitionists seek to reconstruct what they consider to be the corrigible portion of mathematics in accordance with Kantian concepts of being, becoming, intuition, and knowledge. Brouwer, the founder of the movement, held that mathematical objects arise from the a priori forms of the volitions that inform the perception of empirical objects.
https://en.wikipedia.org/wiki/Philosoph ... tuitionism
As I had qualified, "empirical" in his case is linked to the
a priori, not merely
a posteriori.
So in summary [philosophically],
No matter how you try to argue for Platonic Mathematics, it is merely a form of
Mathematical Realism which is fundamentally '
Realism' [
Metaphysical / Philosophical Realism] grounded on Pure Reason which ultimately is illusory.
OTOH, I am countering your Mathematical Realism with the basic of
Metaphysical Anti-Realism [Kantian] which is realistic and grounded on the a priori [empirical].
I am not saying Platonic Mathematics is useless rather its grounding is illusory.
It is on this same illusory platonic grounds that theists conjure their illusory God and reifying such an illusion as real to the extent that such a god would command theists to kill non-theists to the possibility of exterminating the human race [an ultimate concern of Philosophy-proper].
You OTOH is a one-track-path to nowhere and la la land.