# What criteria determines if a proposition is mathematical or empirical?

It seems that there is a distinction between mathematical vs empirical statements.

For example, consider the proposition “All even numbers greater than two are a sum of two prime numbers.” This statement needs to be proven purely in terms of arguments.

However, consider “Workers are more productive if they have a box of jelly beans at their desk”. I could argue that this is the case. However, most behavioral scientists will probably think that an argument is insufficient evidence, and they will ask for the utilization of the [scientific method], possibly by making actual observations.

Consider also the statement “Algorithm A will do task X faster than Algorithm Y”. It is not immediately clear whether this statement is empirical or mathematical. A mathematically minded reviewer might be okay with a purely argumentative reasoning. An empirical minded reviewer would argue that the mathematical reasoning is not sufficient because the process of modeling may have missed something.

In most cases, whether or not something needs empirical evidence is usually pretty clear. However, philosophically, what is the criteria? Are there some cases where the line is blurry?

• Whether workers are more productive with jelly beans is certainly an empirical question. Whether an algorithm is faster is ambiguous; if it is an abstract algorithm on an abstract machine, it is a mathematical question. If it is a program on a physical computer, it is an empirical question. Though, answering the mathematical question is certainly helpful in answering the empirical one. Aug 6 at 15:20
• Mathematics is a set of rules that can be used to describe empirical reality. `1+1=2` is simultaneously a mathematical fact and an empirical fact. Mathematical statements are not empirical (they are metaphysical), but they perfectly fit empirical reality (that is, what is physical, the complement of what is metaphysical). Aug 6 at 17:29
• @RodolfoAP, this seems very much trying to have cake and eat it. The rule- or game-formalist perspective would say that maths is in a substantial sense not metaphysically committing, but is rather only true because of its empirical consequences in the context of our operative scientific theories. If we want to go down the “set of rules” approach then we should embrace that mathematical statements are, in fact, empirically contingent; that’s why the maths fits empirical reality. Aug 6 at 18:08
• There is only one criterion - whether one intends the proposition to apply empirically. Whatever modeling misses is beside the point when the subject is the model itself, and that is mathematics. As Einstein put it:"As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality." Energy conservation is mathematical when it is taken as a logical consequence of mechanical laws, when real systems are taken to obey it - it is no longer mathematical. Aug 6 at 20:13
• I'm having trouble with filling in the blanks: Rationalists tend to ___ and Empiricists tend to ___. Aug 7 at 0:50

First, claims can be both rational and empirical, and mathematical claims are capable of being rational, empirical, or both in nature. So, it needn't be one or the other.

Second, in a sense, all claims are rational and mathematical if you believe, as Quine did, that the basic function of ontological declaration and the use of existential operator is to declare some number if things exist. Any claim also involves a verb which is a relation. "There is a dog present" is mathematical insofar as the premise that the dog is present involves that one dog exists. Of course, a right proper mathematical claim will invoke mathematical language, but the basic subject matter of mathematics is arguably form, shape, direction, quantity, and so on, and so if one can draw a line from the content of the claim to a subject of mathematics, the claim is mathematical. The details of that are subject to the philosophy of math itself, which often highlights not just logic but structure of language.

An empirical statement is one in which the same sort of analysis draws a path back to sensibility in the philosophical sense. Any claim about the world external to our minds essentially qualifies. And claims about dogs are empirical because what a dog is is fundamentally an empirical venture if, like Kant, you observe the basic metaphysical dichotomy between the noumenological and phenomenological.

That being said, not all claims emphasize the rationality, the mathematical, or the empirical to the same degree. If a debate about dogs is being had among philosophers of language in regard what the token 'dog' is, then the claims will be more rational, because the domain of discourse is the mentality of the token 'dog' in language. The semantic content of language is largely a question of the mental. If the debate is about by what physical criteria a dog can be identified by, then the domain of discourse is physical and empirical in nature. This is made obvious by the fact there is a difference between a dog and a 'dog': we use the use-mention distinction in our dialog to discuss dogs as real things and dogs as conceptualized things.

Algorithm A will do task X faster than Algorithm Y.

As someone with some knowledge about physical computation, this claim is rational, mathematical, and empirical. In fact, computer science largely tries to make rational and mathematical sense of empirical facts of physical computation. Algorithm performance, and issues like time and space complexity in computability theory might be discussed in terms of Big-O notation, in which case the debate is mathematical. Or might be discussed in terms of the actual performance of a physical system, such as when a distributed system is built and gigaflops are measured. Obviously, both sets of claims are rational in that they invoke logical consequence. So, robust decision about physical computational characteristics are both rational and empirical. Some call this property dualism, and the philosophy of computer science is concerned with making sense of both the rational and the empirical using measurement.

Claims do not fit in tidy buckets. Language does not fit in tidy buckets. Thought does not fit in tidy buckets. Any attempt to do so is a manifestation of oversimplification.

• Hmm. I think it’s an interesting question whether Quine would agree with the proposal that logical quantification is mathematical. On the one hand, first order logic is logic, and mathematical objects are subject to the primacy of FOL as the foundation of our quantification idiom. On the other, Quine’s theory of membership is itself a distinct predicative set theory, with limitations of description being operative rather than size. So it’s not at all clear whether we should see this as a Platonist move or a Formalist one; if the former, I’d agree, and if latter I’d not. Aug 6 at 17:47
• @PaulRoss If my understanding is right, and Quine rejects mathematics being reducible entirely to logic, then I would presume that is based on the fact that a logical variable itself, before it can even be used in existential quantification, has a cardinality. Given his skepticism of apriorticity, it seems reasonable to conclude he understands that counting is fundamentally an intuitive act and is part of semantic ascent. Given his indispensability argument and his appeal to a natural epistemology, I suspect he would defend that mathematical intuitions are primary to learning about the world.
– J D
Aug 7 at 5:51

In philosophy, the distinction you are talking about would be called a priori vs. a posteriori. A priori knowledge is what is knowable without reference to experience (although not necessarily prior to experience as the name suggests) and a posteriori knowledge is what is only knowable through experience. A priori knowledge includes mathematical knowledge; posteriori knowledge is also called empirical knowledge.

The fundamental test is whether an experiment or empirical observation is needed to answer the question. To answer the evens/primes question, you don't look at physical objects; you don't lay out a series of beans and count them, for example. And, in fact doing so would not even help to answer the question, much less be necessary. No matter how many times you count out a pile of an even number beans and look for a way to divide the pile into two prime piles, you will never know if it's true of all even numbers.

As to your algorithm question, note that algorithms don't have a speed; they have a number of operations based on some parameter of the input data, so one won't be faster than another; it will have fewer operations than another on the same data. Programs have a speed, but a program is not an algorithm; it is an implementation of an algorithm. Figuring out how many steps an algorithm has based on input size is done a priori, so it leads to a priori knowledge. Measuring the runtime of a program is an experiment, so it leads to a posteriori or empirical knowledge. For machines with a predictable time per operation, you could in theory list all of the machine instructions that a particular program will execute and predict the running time, and that would be an a priori operation, just like predicting the orbit of a planet is an a priori operation.

More problematic is this: what if you run a program to create the list of operations it will execute? Is that a priori knowledge or empirical knowledge? My take is this: if the execution is non-deterministic, then the data is empirical. If the execution is deterministic so that you will always get the same list of instructions, and that list is predictable from the program and the input, then it depends on your point of view: you could view the machine as just calculating the result for you, like a calculator might calculate the sum of two large numbers in which case it would be a priori, or you could view the execution as an experiment, in which case it would be empirical knowledge.

• "The fundamental test is whether an experiment or empirical observation is needed to answer the question". This is what I am asking. How do we know if it is needed? Aug 7 at 1:08
• I suppose it's a question of what a New York hotdog would taste like in Pyongyang? Aug 7 at 2:00
• @AgnishomChattopadhyay: We know by context. If we are making a claim in relation to a mathematical system and nothing more, we don't need any empirical data; what we need to do is demonstrate that the claim is true given the rules of that system. On the other hand, if we are making a claim about a physical system (for which we might use mathematics as a language to describe its behaviours), then we would want empirical data. Aug 7 at 2:21
• A computer runs both an algorithm and a program. A program is just a machine-specific linguistic encoding of an algorithm. Opcodes (as binary strings) are no more physical than algorithms. They're both linguistic artifacts, and the construction "the computer is executing a bubble sort" emphasizes the algorithmic nature of instructions (as opposed to heuristics).
– J D
Aug 7 at 5:17
• @AgentSmith I don't understand what you mean by "hot dog in Pyongyang" here. What is the connection? Aug 7 at 11:58

The Modus Ponens (A → B) ∧ A ⊢ B is a logical truth, which means that it does not require any prior assumption to be true (outside the semantic of the terms and symbols involved).

The implication A → B will only be true if you make so suitable prior assumption, for example that A is C ∧ D and B is C ∨ D.

Reference to the real world is just one instance of a prior assumption which will make some statements true and others false. All bats are mammals only because we normally mean bats to be real bats and mammals to be real mammals.

Reference to axioms, directly or through already proven theorems, is another example of a possible prior assumption which will make some statements true and others false.

there is a distinction between mathematical vs empirical statements.

There is and it follows from prior assumptions.

I think the answer that has the most votes, while good in and of itself, does not do justice to your question.

In some ways, the cleavage between "mathematical" statements as you call them, and empirical relates to, as another poster noted, the distinction between analytic (truth-functionally true) and contingent (not truth-functionally true). I'm using these roughly as the logical positivists did, and as they are defined in first-order logic.

A logical theorem, e.g. modus ponens, is analytic/truth-functionally true. As is a proof of Fermat's Theorem, and mathematical proof generally. Your characterization of "mathematical" based on the example you gave, corresponds to what in various ways has been called analytic or necessary propositions. These can uncouple depending on further assumptions. (Kant's definition of analytic would not apply here as it's limited to definitions only).

So to answer your question more robustly: the distinguishing criterion is a theory of deductive proof. Axiomatic systems are systems where given a few primitive assumptions (which themselves require no proof), all further statements, are formally deducible from them. Most accounts of proof, at least orthodox ones, are deductive in this way. This includes mathematical proof.

So this brings us to your last point:

Consider also the statement “Algorithm A will do task X faster than Algorithm Y”. It is not immediately clear whether this statement is empirical or mathematical.

There's a proof that a set of statements are Turing computable called the Church-Turing Thesis. This is a formal proof that has the following empirical implications: if a physical computer meets the minimal threshold of a general computer (Turing completeness), then it will compute at least a subset of Turing computable algorithms.

In other words, Turing computability refers to all the logically possible algorithms. But all the logically possible algorithms cannot be realized empirically. Why? Some empirical laws like entropy limit how much energy can be converted to work, and the availability of energy for conversion to work more generally.

As regards the efficiency of algorithms, my understanding is that you can give formal proofs for the limiting behaviour of a function as it approaches infinity for example. These are purely mathematical/deductive ways that have implications for all empirically possible/instantiated algorithms. However, once you built-in any empirical assumptions into your algorithm, namely about the way the world is, that is no longer a proposition justified by definitions, axioms, and proof alone. Every instantiation of an algorithm is empirical by nature and subject to empirical limits. However, algorithms in the abstract can be stated as proofs.

Note on Quine: I think it's well noted that this distinction as I have characterized it can be attacked, but it's important to begin with it first and expound justifications for it since so much of logic, set theory, mathematics rest on formal deduction. Anything not arrived at through deduction alone, can be said to be empirical.

More problematic is this: what if you run a program to create the list of operations it will execute? Is that a priori knowledge or empirical knowledge? My take is this: if the execution is non-deterministic, then the data is empirical. If the execution is deterministic so that you will always get the same list of instructions, and that list is predictable from the program and the input, then it depends on your point of view: you could view the machine as just calculating the result for you, like a calculator might calculate the sum of two large numbers in which case it would be a priori, or you could view the execution as an experiment, in which case it would be empirical knowledge.

In a deterministic system you still have to plug in the initial conditions. These are empirical. The values to the variables must come from observation. So are the laws of physics: the rules are generalized from observation, despite being exception-less. Now it is true that if the system is actually deterministic, ideally you ought to be able to deduce future states from the initial conditions and the laws of physics alone. However, only the deductive inference is necessary, the premises are empirical. Same with any deductive inference wherein the premises are empirical. I can't deduce that Socrates is mortal without also supposing that every man is mortal and that Socrates is a man, which are a posteriori propositions. So only the deduction itself is a priori, but the premises come from the world. Just by virtue of the fact that the outcome could come out false depending on the truth-value of the premises, you don't have truth-functinoal truth.