Mathematics corresponds to the structure of reality (and omits the qualities that fill the structure).

Mathematics corresponds to the structure of reality — litewave

But, the proponents of the correspondence theory of truth lauded it as composed of logical simples, logical atomism, and even logical monads.

How do you dispell this discrepancy between your assertion and the logical positivists or even reconcile it?

And, kind of hammering the point home, if logical simples, logical atomism, and even logical monads are how we can describe the correspondences of reality through the mind's eye, then does it follow that intuitionism is true, rather than saying that reality is mathematical.

Compare:
1. Intuitionalism is true.
2. Reality is mathematical.

2 is evaluated by 1 at all times, no?

But, then intuitionalism, accordingly to the correspondence theory of truth, is mainly composed of a correspondence of truth bearer's (propositions) being able to mirror state of affairs, situations, tropes, or utilized nowadays truth apt models of reality, yes?

@Banno?

Hello?

The correspondence theory of truth is only part of the story. In common with all expansive theories of truth it's misleading. So there are folk who accept the correspondence theory of truth, and accept that 12/6=2 is true, and hence conclude that there are things to which 12/6=2 corresponds. That's one of the excuses offered for Platonism.

Or you get fluffy stuff like

Mathematics corresponds to the structure of reality. — litewave

Better to treat mathematical statements as grammar. They are descriptions of what we can say if we want to keep what we say coherent.

Hence the string of paralogical and para-mathematical musings in my recent threads. To my eye they lend themselves more to maths as construction rather than discovery.

The correspondence theory of truth is only part of the story. In common with all expansive theories of truth it's misleading. So there are folk who accept the correspondence theory of truth, and accept that 12/6=2 is true, and hence conclude that there are things to which 12/6=2 corresponds. That's one of the excuses offered for Platonism. — Banno

To be honest, I haven't seen many arguments for Platonism with regards to the correspondence theory of truth. It seems to me that the best we got is intuitionalism with regards to mathematics.

However, it seems more pertinent to say that intuitionalism is related in part to logicism with regards to the correspondence theory of truth.

With that sentiment, Hilbert set out to formalize mathematics according to logic, which failed with Godel's Incompleteness Theorem. So, then we have topics like yours about "logical nihilism"; but, no alternative is provided after that.

So, whence does logic end and mathematics begin, could be a short way of asking another person...

But, the proponents of the correspondence theory of truth lauded it as composed of logical simples, logical atomism, and even logical monads. — Shawn

Reality consists of things and relations between them. By "thing" I mean something that is not a relation, nor a structure of relations, so "thing" is something unstructured, indivisible, monadic and therefore a "qualitative stuff" (a quality). Mathematics describes the relations (quantitative, geometric, algebraic... all of these relations can be represented as structures of set membership relations, according to set theory). Mathematics does not describe things (qualities), for example colors, sounds, tastes..., only relations between things, but we have (non-mathematical) words for things too, so we can form propositions about both things and relations and iff these propositions correspond to reality they are true.

Reality consists of things and relations between them. — litewave

So, mathematics best describes these relationships? I would agree. Yet, what's mathematical about hydrogen? Is it a 'thing', as you might say?

Mathematics, it seems, is a constellation of formal syntaxes which, in part, is useful for consistently map-making (re: coherence) and thereby mapping (the) territory with precision (re: correspondance).

So, logicism became linguistic with the advent of the linguistic turn? Do you think this was a reification of thought-through-syntax in mathematics?

I don't understand the questions.

A succinct summary of logical atomism. No longer a popular view.

Mathematics, it seems, is a constellation of formal syntaxes which, in part, is useful for consistently map-making (re: coherence) and thereby mapping (the) territory with precision (re: correspondance). — 180 Proof

What is map-making and why do you call it "coherence"?

If what matters most according to the correspondence theory of truth, is the accurate portrayal of a particular or general 'state of affairs' - through language - of reality... — Shawn

Regarding 'correspondence'.

According to correspondence theory, truth consists in the agreement of our thought with reality. This view seems to conform rather closely to our ordinary common sense usage when we speak of truth. The flaws in the definition arise when we ask what is meant by "agreement" or "correspondence" of ideas and objects, beliefs and facts, thought and reality. In order to test the truth of an idea or belief we must presumably compare it with the reality in some sense.

But In order to make the comparison, we must know what it is that we are comparing, namely, the belief on the one hand and the reality on the other. But if we already know the reality, why do we need to make a comparison? And if we don't know the reality, how can we make a comparison?

Also, the making of the comparison is itself a fact about which we have a belief. We have to believe that the belief about the comparison is true. How do we know that our belief in this agreement is "true"? This leads to an infinite regress, leaving us with no assurance of true belief.

Randall, J. & Buchler, J.; Philosophy: An Introduction. p133 (Cribbed from an old forum post.)

then what does mathematics correspond to in reality according to the mind's eye? — Shawn

I think the original rationalist philosophers argued that, because mathematical truths are known directly, i.e. not mediated by sensory perception, then they qualify as a higher form of knowledge than statements concerning things in the sensory domain, such knowledge always being mediated by the senses.

Aristotle, in De Anima, argued that thinking in general (which includes knowledge as one kind of thinking) cannot be a property of a body; it cannot, as he put it, 'be blended with a body'. This is because in thinking, the intelligible object or form is present in the intellect, and thinking itself is the identification of the intellect with this intelligible. Among other things, this means that you could not think if materialism is true… . Thinking is not something that is, in principle, like sensing or perceiving; this is because thinking is a universalising activity. This is what this means: when you think, you see - mentally see - a form which could not, in principle, be identical with a particular - including a particular neurological element, a circuit, or a state of a circuit, or a synapse, and so on. This is so because the object of thinking is universal, or the mind is operating universally.

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

But it also means that the faculty which sees mathematical facts, is of a higher order than the senses. Which is in many ways preserved to this day in science, for instance Galileo's declaration that 'the book of nature is written in mathematics'. Although there is also the view that current physics has itself become lost in math.

But, at any rate, in terms of history of ideas, the Platonist view is that the intellect (nous) is what is able to grasp the forms and reasons of things, through reason and mathematical intuition. This kind of idea has fallen out of favour in modern thought, due to the predominance of nominalism, which rejects any such conception. But there are still platonists in the modern world, including Kurt Godel, and probably also Roger Penrose.

Map-making is making ... maps (or models). Optimally they are coherent (i.e. all of their properties and functions work together).

So there are folk who accept the correspondence theory of truth, and accept that 12/6=2 is true, and hence conclude that there are things to which 12/6=2 corresponds. That's one of the excuses offered for Platonism. — Banno

Interesting explanation. Probably I am thinking wrongly, but can we put here the Aristotle’s syllogisms? All these principle of “truth” inside mathematics and then, your example, it reminds me about the classic syllogistic method of Darapti, Felapton, Bramantip, and Fesapo.

From a retired mathematician who still dabbles with it, when I work on convergence theory in a dynamical system in the complex plane I always demonstrate theoretical results with computer imagery examples. I am doing that at present, and it is gratifying to watch the sequence of dots approach a fixed point as predicted. That sequence of electronic dots has a kind of "physical" existence but is still in a way non-physical. How does this fit into the current discussion? :cool:

The way math began, if mathematical historians are right, suggests that math subscribes to correspondence theory of truth. Given numbers are abstractions of the natural world, they, in a sense, correspond to an aspect of reality (patterns).

However, at some point math broke free from reality - this happened when mathematicians realized that there really was no need for math to correspond to anything at all. From then onwards, mathematicians began tinkering around with the foundational axioms of math that did correspond to reality and developed entire mathematical universes that have no real-world counterparts to correspond to. Nevertheless, physics seems to be at the forefront of applied math and I'm led to believe that many such mathematical universes seem to, intriguingly, match how reality is i.e. there's a correspondence there!

The way math began, if mathematical historians are right, suggests that math subscribes to correspondence theory of truth. — TheMadFool

I had the idea it was with land title claims and the tallying of agricultural output in Sumeria and Egypt. Land holdings had to be calculated across very irregular shapes, There was a recent discovery about this https://cosmosmagazine.com/science/mathematics/babylonian-tablet-trigonometry-pythagorean-triplets/

So, mathematics best describes these relationships? I would agree. Yet, what's mathematical about hydrogen? Is it a 'thing', as you might say? — Shawn

Yes, the hydrogen atom is a thing with relations to other things, notably to its proton and electron and to the spacetime of which it is a part. Due to its relations it has mathematical properties such as 2 parts (proton and electron, which both have their own parts, even the electron because if the electron had no parts it would be an empty set and an empty set does not have properties such as mass or electric charge), spatial size and shape, extention in the time dimension (lifetime), value of mass/energy...

To clarify my ontology, every concrete thing is a collection of parts (the smallest collections are empty collections/sets, that is non-composite concrete objects). It is important to note that while a collection has a structure, this structure is constituted by the relations of the collection to its parts, and none of its parts is identical to the collection. So the collection as a whole is a thing too, different from its parts, and this thing is something unstructured that stands in relations to other things, notably to things that are its parts.

All possible collections are rigorously defined by set theory, which can represent all mathematical properties as collections (sets).

I had the idea it was with land title claims and the tallying of agricultural output in Sumeria and Egypt. Land holdings had to be calculated across very irregular shapes, There was a recent discovery about this https://cosmosmagazine.com/science/mathematics/babylonian-tablet-trigonometry-pythagorean-triplets/ — Wayfarer

Thanks for the link. I'll read it and get back to you if I find anything interesting.

the smallest collections are empty collections/sets, that is non-composite concrete objects). — litewave

Do you mean atoms?

I don't claim that truths of propositions that are joined into a longer proposition are necessarily independent from each other. To logically prove whether or not they are independent we would need to analyze the things and relations they refer to, down to the lowest level if necessary (to the smallest parts, in the case of concrete things).

No, atoms in physics are obviously not non-composite things.

But the original meaning of atom was literally that. Atom meant non-divisible or non-composite. The atom in modern physics doesn’t mean that, but your ‘non-composite concrete objects’ are pretty well exactly what the atom was understood to mean when the term was coined.

Consider mathematics, like any form of language, to be a tool. As such, the part of reality which it must correspond with is the part which consists of means and ends, intentions,, and fulfilling them, described by final cause, purpose, and function.

As rightly points out, "reality" is not something which we have a firm grasp of (though many like to deny this fact), so a judgement of correspondence is never a simple issue. This part of reality, which consists of final causes, means and ends, intentions, purposes, functions, etc., we barely even recognize as being a part of reality.

From then onwards, mathematicians began tinkering around with the foundational axioms of math that did correspond to reality and developed entire mathematical universes that have no real-world counterparts to correspond to. Nevertheless, physics seems to be at the forefront of applied math and I'm led to believe that many such mathematical universes seem to, intriguingly, match how reality is i.e. there's a correspondence there! — TheMadFool

Even though many descriptions of a universe by mathematicians don't correspond to our universe, they correspond to other possible universes. And what is the ontological (existential) difference between a possible universe and a "real" universe? I think none, so all possible universes exist and descriptions of all possible universes correspond to reality. There is no difference between correspondence theory of truth and coherence theory of truth.

But the original meaning of atom was literally that. Atom meant non-divisible or non-composite. The atom in modern physics doesn’t mean that, but your ‘non-composite concrete objects’ are pretty well exactly what the atom was understood to mean when the term was coined. — Wayfarer

By 'non-composite concrete objects' I mean empty sets, which have no parts by definition. No amount of empirical evidence can prove than an empty set has parts.

But an empty set is nevertheless a concrete object?

Or, should that be the empty set, as there can’t be more than one, can there?

Even though many descriptions of a universe by mathematicians don't correspond to our universe, they correspond to other possible universes. And what is the ontological (existential) difference between a possible universe and a "real" universe? I think none, so all possible universes exist and descriptions of all possible universes correspond to reality. There is no difference between correspondence theory of truth and coherence theory of truth. — litewave

My reading of the correspondence theory of truth requires two essential components:

1. An actual reality. Call this R
2. A proposition about that actual reality. Call this P

When P matches R, there's a correspondence and then we can claim P is true.

If you wish to include possible worlds/realities, my advice would be to coin a new word and for the match between propositions and such worlds to avoid confusion.

Of course there's the matter of whether or not every possible world is actual (modal realism) or not but, speaking for myself, I'd like to retain the distinction between possible and actual. It's useful not to believe some things are actual.