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When I'm thinking about mathematics, I usually imagine that every sentence in the language of arithmetic is either true or false, in reality. Thus, I imagine that truth comes first. Afterwards come axiom systems that are intended to prove as many true sentences as possible, while not proving any false sentences.

However, in practice we seem to begin with a formal system (that is, we begin with provability), and only afterwards do we define truth. For example, we might start with a formal system, say ZFC, and use it to define N, the set of natural numbers, and then prove that the Peano axioms are satisfied by N; that is, they're true (with respect to N). But notice that we had to begin with a formal system, namely ZFC, before we could talk about truth.

As a result, I am philosophically confused about which comes first, truth or provability. This has practical ramifications. For instance, suppose I wanted to write an introductory book about the foundations of mathematics. Which would I talk about first?

What are the major positions in this debate, and where can I learn more? Note that I am not specifically interested in whether or not mathematical entities have any real and independent "existence" except as a means to working out a "time ordering" or "logical ordering" on truth and provability.

EDIT. I'd like to point something out. In mathematical logic, there's an enormous difference between truth and provability. Truth is a relationship between models and sentences, while provability is a relation between axiom systems (or more accurately, "formal systems") and sentences.

So for example, we can say, "For the standard model of arithmetic, Goodstein's theorem is true." Since this has been proven, it is believed true. We can also say, "From the axiom system known as Peano Arithmetic, Goodstein's theorem is provable." Since this has been disproven, it is believed false.

So when I say, "Which comes first, truth or provability?" what I really mean is, "Which comes first, models or formal systems?"

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    +1 Good question! My qualm: "Which comes first" is not really well-defined. If by time-ordering you mean historical development, then it is clear that many, but not all mathematical statements where seen to be true before they were proved (think of "1+1=2", Pythagoras' theorem, etc.). And what might "logical ordering" mean? My suggestion: You may want to rephrase the question in general terms of "relation between truth and provability". Or does this deviate too much from your actual concern? – DBK Apr 25 '13 at 6:59
  • @DBK, I tried clarifying my actual concern with a practical question: "As a result, I am philosophically confused about which comes first, truth or provability. For instance, suppose I wanted to write an introductory book about the foundations of mathematics. Which would I talk about first, truth or provability?" Does this address your qualm? If not, feel free to edit the question as you see fit. – goblin Apr 25 '13 at 7:19
  • Right, had skipped that bit somehow. My bad. (Still not sure how the didactical perspective would necessarily illuminate the mathematical question, but I'm sure someone else will provide a good answer.) – DBK Apr 25 '13 at 11:34
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    Truth without provability is random and provability without truth is without objective. – Annotations Apr 25 '13 at 12:24
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    In front of me, I have a copy of Shapiro's Thinking about Mathematics: The Philosophy of Mathematics. Looking at the table of contents, Truth (in "2.2 Global Matters") comes before Theorems (in "2.4 Local Matters"). – user3164 Apr 25 '13 at 18:18
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If you're talking about mathematics, or other abstract subjects, it is not clear to me that there is a satisfying answer to your question unless you broaden your notion of what mathematics consists of to include non-rigorous pattern recognition.

What does one mean by "truth" in mathematics? The basic usefulness of arithmetic comes from the fact that it is essentially a simple theory of matter, used to describe discrete objects which under normal manipulation maintain their integrity and distinctness from other objects. But you cannot prove that 2 + 2 = 4 by shuffling around apples; you can only motivate the notion that addition (aggregating collections) is a useful concept when counting, and that it should be defined in such a way that 2 + 2 = 4; additional evidence for the wisdom of this convention is provided by showing that it generalizes to many other contexts.

Is it true that 2 + 2 = 4? Yes: because we have defined addition so that it is so; and we have done it to describe how to aggregate collections. As to the "obviousness" that when you bring two collections of 2 together you will obtain 4 objects, this is essentially a recognition that matter behaves a certain way when you move it around in space, under certain conditions — if you throw together the two collections at one another (or a hard obstacle such as a wall) at high speed, you don't obtain four apples so much as a variable amount of apple pulp. If this is "not what we mean by addition", then does that mean that the truth of 2 + 2 = 4 is contingent on the particular physical process of bringing things together — that you would get four, if only you treated the apples more gently? I find it a little hard to escape that notion. Of course, we find arithmetic useful beyond the gentle transportation of produce to such applications as building bridges and lasers, but this is because we are brave and assume that our tools for counting could be easily used for more general problems. This turned out to be right in practise, provided we are willing at least to extend to using fractions and negative numbers; but this probably owes more to the fact that arithmetic has enormous expressive power (for instance, being able to model any formal syntactical system by Gödel enumeration) and internal consitency (in that two mutually exclusive consequences don't seem to be easy to produce), rather than the inherent truth of its propositions.

Given that we've generalized well beyond its original motivating examples, does the truth of 2 + 2 = 4 only emerge from a proof? Well, you can define addition in a system such as Peano Arithmetic, and thereby prove that 2 + 2 = 4 from the primitive definitions, but this is an anachronism when talking about whether 2 + 2 = 4 was established by proving it. It seems to me that people formulated the idea of addition through observation of regularity in the world, so that "2 + 2 = 4" had the same sort of quality as other observations of pattern, such as "the sun rises in the east" and "water is wet". They were empirical recognitions of matters of fact about the world, but served more to give definition to the functional roles of 'plus', 'east', and 'wet' than to describe the properties of the substances involved. By identifying and defining the concepts, it reinforced the way in which they could think of the world.

Mathematical "facts" are descriptions of patterns, which we note are instantiated in the world. They are not so much true propositions, as they are useful patterns, whereas we find that competing propositions such as "2 + 2 = 5" don't really fulfill any useful function. In this sense, you might say that truth has come before provability for a statement such as "2 + 2 = 4", because while not universally applicable, it is a concept which has such versatility that we may find it in effect "supported" again and again in our interaction with the world.

In the sense that any particular mathematical statement models something which can be easily described in the world, we may say that its truth (by demonstration of positive examples of its use as an idea) has preceded the notion of provability. Proof then seems to arise as the art of substituting thought-experiment for actual experiment — with the attendant convenience of not having to demonstrate propositions physically, but similarly with a risk of escaping realizability entirely.

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I believe truth is the mother of all logical systems. Even undecidability is based on truth, propositions we prove we cannot demonstrate they are true or false, which are new potential axioms in fact as the axiom of choice which seems so obvious : it is always possible to select exactly one element of each set of a collection of sets. In fact we always instinctively try to prove that something is true or false, right or wrong and this is true for mathematics. At the beginning people did not want to prove things which seemed obvious, there were true as euclidian axioms were. It is a relatively new idea to consider the proof as central, it's maybe the main difference between modern mathematics and physics, in physics what really matters is truth, what can be tested. Truth is deeply rooted in our culture, much more than provability.

  • I want to agree, but the way that rigorous math is actually done seems to define truth in terms of formal systems. So it would seem that, in any rigorous work, provability comes first. (Which bothers me.) – goblin Apr 30 '13 at 1:58
  • Yes, you are probably true but from an historical point of view truth comes first without any doubt. Concerning the present I think physicians are sort of mathematicians, for instance the impulse on spectral theory comes from quantic physic. Mathematics is too important to restrict it to what mathematicians do, very often intuitive ideas coming from other fields fertilize maths and what it is behind them is truth not provability. For instance, the collapse of the Rutherford model of atoms comes from mathematical considerations but the quantum theory comes from intuition. – Mauceric May 1 '13 at 9:16
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What is truth?

Characterizing a sentence as true adds nothing new to its content, for ‘It is true that 5 is a prime number’ says exactly the same as just ‘5 is a prime number’. The adjective ‘true’ is redundant and is not a real predicate expressing a real property such as the predicates ‘white’ or ‘prime’ which cannot be eliminated from a sentence without an essential loss for its content (Frege).This idea gave an impetus to the deflationary theory of truth. There is no such property as truth and thus there is no need for, or sense to, a theory of truth distinct from a theory of truth ascriptions. Truth is not taken to be explicitly defined, but rather the truth conditions of sentences are taken to be described.

I usually imagine that every sentence in the language of arithmetic is either true or false, "in reality". I am not specifically interested in whether or not mathematical entities have any real and independent "existence".

I am not sure what you want to mean. But I invite to think at first about truth in the real world, about “indubitable” self-evident truth.

Why think that a priori justification implies that this sort of justification is entitled to ignore empirical information ? Why couldn't a priori justification be defeated by empirical, not just a priori, considerations? Something like this has actually happened, Kant was a priori justified in believing that every event has a cause but, because of developments in sub-atomic physics, we are not. The principle of sufficient reason is a synthetic a priori and can be "defeated" by another model. The principle is only justified in the framework of a deterministic conception of nature, and contemporary physics does not any more support. In a radioactive particle decay, it is indeterminate if decay or not becomes at time t. The behavior of radioactive particles constitutes a counterexample to the version, as Hume uses, of the principle of sufficient reason: No event, of whatever type, can happen at time t without something determining its occurrence at that instant. The principle of sufficient reason is example of a fake “indubitable“ self-evident truth. The Greeks were a priori justified in accepting Euclidean geometry but we are not because of developments in cosmology. The net effect of Einstein’s use of non-Euclidean geometry in a physical theory was to disestablish the view that mathematics is a source of a priori knowledge about the empirical world. Euclidean geometry then is another example of a fake “indubitable” self-evident cosmological truth. A priori justification allows that experience might defeat a priori justification.

One primary dispute is over the source of the a priori knowledge. On one standard interpretation non-inferential, a priori justification is solely justification based on understanding the proposition at issue.The problem in linking a priori warrant to self-evidence is that a priori warrant is compatible with inferential warrant, wherein a proposition owes its warrant to inferential relations with other propositions, as might a theorem in a mathematical system but theorems in mathematics are not always self-evident. It is not obvious that a priori warrant for a proposition requires epistemic indubitability of this proposition. A priori justification for a proposition apparently can be subject to epistemic defeat. Truth's ascription conduciveness is a necessary condition for epistemic justification. The claim that a source of beliefs is truth's ascription conducive is a contingent empirical claim that need be supported by empirical investigation.

Outside the real world, in the abstract world of pure mathematics, what can mean truth without provability? What is the sound of one hand clapping?

Which comes first - truth or provability?

According to the deflationary theory of truth, characterizing a sentence as true adds nothing new to its content,the adjective ‘true’ is redundant and is not a real predicate expressing a real property. In physics there is no evidence available in principle that can distinguish a theory’s truth from its utility and reliability in prediction. In mathematics truth is what was proved from axioms. Can truth be stated justifiably, without first having theory of truth ascriptions? Is there a provability-independent way to reconstruct phrases like "the truth really there"? My answer: No.

  • Relative to the deflationary theory of truth, we may then rephrase the question as: "can statements be stated justifiably, without first having proof?" – Niel de Beaudrap May 2 '13 at 14:24
  • @NieldeBeaudrap Grammar of utterances suggests that we use them to predicate a property, truth, of sentences or propositions; but the deflationary theories all deny this. There is no such property as truth and thus there is no need for, or sense to, a theory of truth distinct from a theory of truth ascriptions. For deflationists, the classical theories of truth, such as correspondence, coherence and pragmatic, are wrong-headed from the start, for they are attempting to analyze something which simply there is not there, there is no such property. – Annotations May 2 '13 at 16:11
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    @NieldeBeaudrap Theories of truth must be really theories of truth ascriptions, they cannot be theories of what truth is. We may then rephrase the question as: "Can statements be stated justifiably, without first having theory of truth ascriptions? My answer: No, otherwise is “truth” in rhetoric or poetic sense. – Annotations May 2 '13 at 16:12
  • How then would you address the notion of mathematical truths, outside of formal systems? Before anyone developed formal systems, were statements such as "2+2=4" ungrounded? Or do you mean to say that the move to restrict 'proof' to formal systems is itself misguided? (Please excuse the comment which I removed when I resolved my own confusion about your comments.) – Niel de Beaudrap May 2 '13 at 16:48
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    @RicardoBevilaqua, you write: "In mathematics truth is what was proved from axioms." I think there's more going on than you presume. For example, if I write you down some axioms in the language of arithmetic which prove "there exists x and y such that x+y is not equal to y+x" then we would both agree that this sentence is false relative to the standard model of arithmetic, despite that we have proved the sentence from the axioms. Therefore, although the axioms may be consistent, they do not capture that which is true for N. – goblin May 3 '13 at 5:11
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Why is 1 less than 2 ?

What is difference between less and more anyways ?

You see, reasoning for and proving these primitive notions of logic to be logically true, or further breaking them down, will tend to make your brain melt. To prevent this melt down (this reverse look-up for ultimate truth, the infinite why?), these primitives instead, are treated as generally accepted true enough basics. Prerequisites of intelligence. Like a platform to build upon, an un-provable platform.

So, yes, there stands a base solid enough (or true enough), beneath all the provable and the derived truths achieved in Mathematics. But if you try to place provability before this base's acceptance, you won't be able to get started at all.

And I think Gödel also said something about a part of system being true but not provable.

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A: "P is True." B: "Prove it!"

A proves it.

A: "You see? It WAS true!"

P's provability has allowed A to prove P. P had been true all along.

Had P been provable all along, too? Not necessarily. P's "Provability" can only have meaning in relation to, say, A or whoever tries to prove it. It is not inherent to P.

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    This involves a commitment to a particular approach to the notion of truth. If the truth status of P is meant to be seen as an absolute of some sort, then saying that it "was" true is misleading; it simply "is" true as a static fact (we wouldn't usually take seriously the question of whether 2 + 2 was 4). The proof is a means to uncovering a feature of a model, taken as a substitute for the world, so that we speak conventionally of things being true of the model the way we would speak of things being true of the world, where we suppose that the existence of the latter is independent of us. – Niel de Beaudrap May 2 '13 at 12:33
  • A may say "It WAS true" even when the truth status of P is meant to be absolute. What A means is:"I WAS right. P IS true. " – user1975053 May 2 '13 at 16:27
  • Nevertheless, the claim by A is unjustified, unless A actually had a proof (or equivalent justification of it) already. So then is A "right", or just that they made a lucky guess? If I make a weather prediction on the basis of a coin flip, can I be meaningfully "right"? – Niel de Beaudrap May 2 '13 at 16:36
  • Of course! Guessing is just another way to truth. Good old Fermat was probably just guessing when he stated his proverbial theorem, which was true then as it now is, but not provable (to Fermat). – user1975053 May 2 '13 at 17:17
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    In that case, your position is that truth precedes proof because truth precedes everything. You would be better off just stating that; though my downvote would stand. – Niel de Beaudrap May 2 '13 at 17:38
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It's neither. They were discovered simultaneously.

In the specific case of mathematics one only has to glance at the history of mathematics to see this is the case. Mathematics does not start with the ZFC axioms - this was part of a campaign to ground all of mathematics in sets and logic. Mathematics contrary to what is often taught in most 'history of mathematics' did not start with the Greeks. They expanded its range and technical resources.

Mathematics started with arithmetic in its most basic form - addition. There it is so easily proven that 3=2+1=1+2 that it requires no proof; and simultaneously they are arithmetic truths.

Of course this not proof in the modern sense of the word, or even in Euclids; but then what is proof and what counts as proof is an idea that is and has undergone historical change. The same is true for truth.

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In finite systems it is very simple. For example, is Goodstein's theorem true? If it is false, we can specify a $n$ such that is Goodstein sequence terminates. Otherwise, it is true. This is clearly either true or false, even if our axiom system isn't powerful to prove it.

A somewhat easier example would the consistency of Peano Arithmetic. If it is inconsistent, we can write a proof that 0=1. Otherwise, we can not.

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Truth and provability are different flavours of the axioms you use to deduce things about axioms. Truth-preserving manipulation is Classical logic, Proof-preserving manipulation is Ituitionistic logic.

It is all about what logical inferences preserve. You can construct formal reasoning systems to preserve pretty much any property from statement to statement. Be it truth, proof, resources, degree of certanity, and even combinations thereof.

  • How does one "preserve" a proof? – Niel de Beaudrap Apr 27 '13 at 2:40
  • @Niel, i believe one does not preserve a proof, but rather, provability. – goblin Apr 27 '13 at 5:07
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    And how does one formulate a formal logic in which the rules of inference do not preserve provability? – Niel de Beaudrap Apr 27 '13 at 11:14
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    @NieldeBeaudrap, good question. Karl? Help us out here mate. For instance, why does double negative elimination (which is not a valid inference rule of intuitionistic logic) fail to preserve provability? – goblin Apr 27 '13 at 12:15
  • @NieldeBeaudrap You take the axioms and inference rules of Inductive (provability preserving) Logic, and remove one ore more of them. – Karl Damgaard Asmussen Jun 21 '13 at 2:08

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