Correspondence theory of truth and mathematics.
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, and therefore what can be platonically described as the mind's eye, then what does mathematics correspond to in reality according to the mind's eye?
Comments (66)
Mathematics corresponds to the structure of reality (and omits the qualities that fill the structure).
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?
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?
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 Quoting 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.
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...
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.
So, mathematics best describes these relationships? I would agree. Yet, what's mathematical about hydrogen? Is it a 'thing', as you might say?
So, logicism became linguistic with the advent of the linguistic turn? Do you think this was a reification of thought-through-syntax in mathematics?
What is map-making and why do you call it "coherence"?
Regarding 'correspondence'.
Randall, J. & Buchler, J.; Philosophy: An Introduction. p133 (Cribbed from an old forum post.)
Quoting 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.
[quote=Lloyd Gerson, Platonism vs Naturalism] 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. [/quote]
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.
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.
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!
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/
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).
Thanks for the link. I'll read it and get back to you if I find anything interesting.
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.
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.
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.
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.
Or, should that be the empty set, as there can’t be more than one, can there?
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.
By a concrete object I mean any collection, so an empty collection would be the simplest concrete object.
I know that sets (collections) are often regarded as abstract objects, but in those cases a set is meant as a concept/property (which is instantiated in concrete sets). A concept/property is an object that is not a collection but it has instances in collections.
I understand, I just meant to point out that if all possible (logically consistent/coherent) universes are equally real as the one we live in, correspondence theory of truth becomes identical to coherence theory of truth.
How?
Every consistent description of a world corresponds to a real world.
[quote=cosmosmagazine.com]To simplify the discussion, let’s use a right-angled triangle with shorter perpendicular side s, longer perpendicular side l, and diagonal d, such that s² + l² = d² .
Columns two and three of Plimpton 322 simply contain values for s and d respectively for the series of Pythagorean triples. Column four is just a list of the numbers 1 to 15, so we can remember which row we’re up to. But column one represents the ratio d² / l², and since we’re given the value of d in column three, we can calculate l, and voila … a complete Pythagorean triple (s,l,d) is revealed![/quote]
3, 4, and 5 are a pythagorean triple: 3² + 4² = 5²
The yield of the land corresponds to its area.
Imagine you had 9 square units of land (3²).
Suppose now, that new land is cultivated and the ratio of the total yield to that of the new land cultivated is 1.5625 (roughly one and a half times). This is basically the ratio of the total area of land that's cultivated to the area of the new land cultivated = d²/l².
We can now calculate l = sqrt(d²/1.5625) = 4 units. The new land cultivated can be thought of as a square 16 square units with sides 4 units. The area of land you began with (3²) + the area of the new land cultivated (4²) = Total land cultivated (5²). From how the yield scaled up (× 1.5625), we could determine the area and dimension of the new land cultivated. I dunno!
Isn't that begging the question? By the way if a world has to be qualified with real as you do in "...a real world", it suggests that worlds can be unreal. Care to expand and elaborate.
As I said, I think that all possible worlds are just as real as our world because I don't see any ontological difference between possible and real worlds.
Commitment to the correspondence theory means commitment to a model's actual existence: properties, relations and all. (Platonism.)
The same commitment isn't required in order to do model theory, because models, like all mathematical entities, might be fictions, like Santa Claus.
Neither is it required in order to do nominalism (reference theory), and examine the correspondences (albeit conventional or pretended) between words and things, or other words. In order to take, that is to say, a mathematical or literary or pictorial story and examine its pretended connections to existing things or events (e.g. world war II) or, that perhaps not being an option, to other words and pictures (numerals, number lines, Santa pics, real old man pics, etc).
Quoting Shawn
Yes, I think so...
Quoting Shawn
No idea what you mean, although actual existence of properties etc is what an anti-platonist can't handle. Is my understanding of 'platonic'. So if 'minds eye' means imaginary... No I can't parse it.
Quoting jgill
Hi there from an ignoramus.
Quoting jgill
I would offer: the non-physical aspect is the pretended or conventional reference (by the dots, in sequence). That leaves it open to analyse the reference as fictive, like a Santa story, or factual, like a history. Either way, there is no need to infer reference to non-physical entities. If you don't want to... Do you?
PS why the scare quotes?
Modal Realism! I quite like it that there are more worlds out there populated by unicorns, fairies, angels but what scares me are vampires, ghosts, werewolves, zombies.
What's impossible to you?
Logically inconsistently defined objects. Objects that are not what they are. Objects that have properties they don't have.
It may not be obvious whether an object is consistently defined, because its definition, its properties include all its relations to all other objects in reality, so it must be defined consistently in relation to everything else. But at least when you interact with an object, you can know that it is consistently defined without having to check consistency of its relations to everything else, because if you interact with it it must exist and inconsistent objects cannot exist.
Perhaps surprisingly, whatever you are doing at this moment, it is impossible for you not to be doing it, at this moment. Simply because it would be a contradiction, an inconsistently defined event, if you were not doing what you are doing. For a copy of you in a different possible world it might be possible not to be doing what you are doing but not for you.
Quoting Shawn
Hmm. Thought I replied to this yesterday, but must not have clicked "Post"...
If physics is applied mathematics, then mathematics is applied logic...?
But what is to count as a simple, as the atom from which you derive the world? Whatever you choose will be arbitrary - we might choose otherwise.
Go back to your first post:
Quoting litewave
...Can you provide an indubitable account of what that "correspondence" consists in?
That's the core problem for correspondence.
Seriously?
If this were so, then since in some possible world you didn't write that post; and since all possible universes exist and descriptions of all possible universes correspond to reality, you really didn't wright that post.
How will you avoid such inconsistency?
You have set the scope of "...exists" across all possible world instead of within the scope of each possible world, and that results in inconsistency.
That already goes too far; you would next be asked to explain the nature of that "match" of the "correspondence". That's the fatal flaw of correspondence theory.
As @Wayfarer impied, what is concrete about an empty set?
Quoting litewave
This is at odds with extensional logic, in which a property is a collection of objects; so "...is red" is the collection of red things.
But then all you have done is claim that anything could be true.
The point is surely to sort out the way things actually are from the way things might be. If every possible world corresponds to the real world, you have no way to do this.
The smallest parts, empty sets, are obviously "simples" in the sense that they have no parts. But any collection is also a "simple" in the sense that the collection as a whole is an indivisible/unstructured thing that stands in parthood relations to other things that are its parts.
Quoting Banno
A proposition describes an object by affirming that the object has certain properties. If the object really has those properties then the proposition is true - that's how a proposition corresponds to reality.
Example: Proposition "Planet Earth has approximately a spherical shape with a radius of 6,370 km" is true and thus corresponds to reality iff planet Earth has approximately a spherical shape with a radius of 6,370 km. And the properties attributed to Earth in this proposition are relational (geometric/quantitative) and thus mathematical.
...and yet to understand empty sets one needs all the paraphernalia of set theory. SO if they are to form the "simples" of a logical system, it is only by presuposing set theory. Which is not all that simple.
Quoting litewave
Yes! Anything can count as a simple.
Quoting litewave
..and? That does not explain the "correspondence" in the correspondence theory of truth. Indeed, while correspondence is about what is the case, you've moved to affirmation, which is distinct, and quite different. One can after all affirm things that are not true.
So I don't see this approach as getting very far.
Assuming that it is really "me" who lives in different possible worlds (which I don't think is a correct definition of "me", since my consciousness is clearly limited to only one of those worlds), I can say that I wrote the post in this world but not in some other worlds and I can also say that I wrote the post in reality as a whole, which is the collection of all possible worlds. Similar to saying that I watched Citizen Kane in Germany but I did not watch Citizen Kane in France, and I watched Citizen Kane in reality as a whole. This way inconsistency is avoided.
We agree that it is true in this possible world that you wrote the post; it is not true in some other possible world?
Quoting litewave
So do you think mathematical statements are true in this possible world because they are true in some possible world?
Then if it is true that in some possible world you dd not write that post, wouldn't it be true in this possible world, too?
Arn't mathematical statements true in all possible worlds?
That it is a collection, rather than a property.
Quoting Banno
I don't think that a property is a collection. Redness is not the collection of all red things but something that is had by all red things. The red things are the extension of the collection of red things as its parts, but they can also be said to be the extension the property redness as its instances. If you think that properties are collections then reality consists only of collections, which are concrete things, because properties as abstract things that have instances don't exist.
Anything that is consistently defined and thus identical to itself.
Quoting Banno
It is useful to sort out the way things are in our world from the way things are in other possible worlds.
That's the very definition of a property in first-order logic. First order logic is extensional by design.
So you using a non-standard interpretation?
Quoting litewave
SO for any proposition P you have:
P is true IFF it is consistent and identical with itself
Consistent with what? "Lightwave wrote this post" is consistent, but not true - I wrote this post.
Quoting litewave
...appears incompatible with...
Quoting litewave
That two things correspond is a judgement. Correspondence is never anything more than a judgement. So there's really no such thing as "when P matches R", just the judgement, and the claim.
Set theory is based on the simple and self-evident fact that objects constitute a collection. A collection can be defined by listing all its parts or by specifying their common property. The problem with defining a collection by the common property of its parts is that such a definition may be inconsistent, so this kind of definition has been narrowed by certain axioms that select certain kinds of collections. It doesn't mean that some axiomatizations of set theory are correct and others wrong; they just select different kinds of collections.
Quoting Banno
Affirmation is in the nature of propositions. Proposition is a tool of communication, which by affirming something provides information. If a proposition affirms that a certain object has a certain property and the object in reality does not have the property, then the proposition does not correspond to reality and thus is not true.
It is not true that in some other possible world I wrote the post.
Quoting Banno
Only if one of those possible worlds is this world.
Quoting Banno
It would be true in this world that I did not write the post in a different world in which I did not write it.
Some are, for example "1+1=2" (I hope). Some are not, for example "Sum of interior angles of a triangle is always 180°."
As far as I know, property in first-order logic is regarded as something that can be had or satisfied by an individual and it is not necessary to interpret property as a collection.
This proposition describes me (Litewave) inconsistently by referring to me, a person who doesn't have the property of "having written this post", and affirming that I have the property of "having written this post". It is as if you wrote that "Someone who didn't write this post wrote this post", or "A circle is not a circle". By affirming that I have and don't have the same property, the proposition is inconsistent and therefore does not correspond to reality. It defines me as a thing that is not identical to itself and such a thing cannot exist.
And a few of the things you have said seem clearly incorrect.
But all this is a side issue for the main thread, and I'm no logic tutor, so let it pass.
https://youtu.be/RUzbmIKVAHo?t=47 :wink:
Quoting Goodman, p49
When a witness in a trial swears to tell the truth, whole truth, and nothing but the truth? Basically this means that the statements (spoken, written, sign language, etc) describe events in the real world as accurately as the witness is capable of doing.