### On Proof and Evidence

Words, words

Sometimes – actually oftentimes – we can get pretty sloppy and careless in our use of words.

Take the use of the words “proof” and “evidence”. Proof and evidence, like speed and velocity, or theory and guess, have colloquial definitions that often lead to confusion. In order to smooth the progress of communication and avoid misunderstanding, these words have been given technical definitions in science and philosophy. For example, speed is defined as the magnitude of velocity; the latter is a vector, the former is the scalar magnitude of that vector. Also, a scientific theory is not simply a guess; rather, it is a system of ideas constructed from a verified set of generalizations and observations. In the same way, scientists and philosophers use the words proof and evidence to designate two very different things. For example, we prove a mathematical theorem instead of “finding evidences” for its truth, while we accumulate the evidence for a particular scientific theory but we never “prove” a theory.

What’s the difference? The distinction is best illustrated by examples.

The calculus of pebbles: a rocky example

The fist example illustrates the nature of scientific evidence. Suppose you are walking on a beach when you decide to pick up exactly two pebbles and put them in your pocket. You then walk a few paces forward before picking up three more pebbles and putting them in the same pocket you put the first two pebbles. Again, you walk several paces forward before you decide to sit on a boulder by the shore. You fish for the pebbles in your pocket and count how many there are, and you find that there are five pebbles all in all. Since you remember that there were initially no pebbles inside your pocket before you picked up the first two, you are led to the hypothesis that two pebbles plus three pebbles will give you five pebbles. You throw the five pebbles and go home. The next day you decide to perform the same experiment: you walk along the same shore as the day before; you pick up two pebbles and put them in your initially empty pocket; you walk a little bit; you pick up three pebbles and put them in your pocket; you then walk a little bit, sit down on the boulder and finally count how many pebbles you end up with. Again, you find that you end up with five pebbles. You do this thing every day for a whole week, even a month, and each time you performed the same damned experiment, the result always confirms your hypothesis that two pebbles plus three pebbles equals five pebbles. You then decide to make the following scientific generalization based on your findings: “Two pebbles plus three pebbles always equals five pebbles.”

This generalization was inspired by the evidence for its truth. But note the ‘always’ that appears in the generalization. This ‘always’ is what makes the generalization risky, and as a consequence interesting and useful. It is interesting because it tells something about the workings of the universe (about the way pebbles add in our universe, for example). And it is useful because whenever an observation seems to contradict this generalization, an interesting problem arises, problems such as: “Why did I end up with one less pebble? Perhaps there’s a hole in my pocket.” You check your pocket and lo and behold, you find the hole that explains the missing pebble.

Notice that the more confirmations of the generalization you get, the more evidence you have of its truth. Notice also that the convincingness of the evidence is increased when it is obtained by someone else other than you and when the beach from which the evidence is obtained is a very different kind of beach from the beach that first inspired the generalization. Using these much more convincing evidence one can then formulate the broader generalization: “Two pebbles plus three pebbles always and everywhere gives five pebbles.” This last generalization is obviously riskier, but is also more interesting and more useful than the former.

Now, after perhaps a thousand such experiments, you get sick and tired of repeating the same wretched experiment that gives the same results over and over again that you say to yourself, “Well, since my generalization seems to be valid always and everywhere, let me give it a more imposing name, say ‘The Law of Pebble Addition’, and let me leave it at that. I have more important things to do with my life than counting pebbles.”

A very wise decision; I would recommend this course of action to anyone. However, the following questions can be asked with justice: How certain are you of the truth of your Law of Pebble Addition? How can you be absolutely sure of its validity, generality and universality? Does it apply today in the Andromeda Galaxy, or in the farthest part of the universe where pebbles are found? Did it apply back in the Precambrian Period, or will it apply five billion years into the future?

If you are honest and careful enough, your answer would be something like this: “No, I am not certain of its truth, because although I have a lot of evidence of it being true, the body evidence is nonetheless finite. I have never been to the Andromeda Galaxy, let alone the farthest part of the universe where pebbles are found. Also, I was not alive during the Precambrian, and I will not be alive five billion years into the future. Consequently, I cannot be certain about the generality and universality of the Law of Pebble Addition.” Of course you should add, “But I have a mountain of evidence that I can marshal to support my Law. And if you want, you can test it in places where I haven’t, perhaps using procedures I haven’t used. That way, you can convince yourself that the Law holds good always and everywhere, although I don’t recommend it to you, since I’ve already done the experiments and I think it’s a waste of your time if you repeat it. But that’s your call.”

We see from the example given above that evidences are observations that support a particular hypothesis. An empirical generalization or hypothesis, such as the Law of Pebble Addition, is inferred from the given body of evidence using the process of induction. Induction or inductive reasoning works in such a way that the more evidence you can gather to support a particular hypothesis, the greater becomes the probability of that hypothesis being true, which means the more you can bet on it. Some empirical generalizations, such as the conservation of energy, the conservation of angular momentum and the principle of invariance, are probable to the degree that they are often times called “Laws of Nature” (take note of the grave capital letters). Calling them Laws of Nature simply means that they are very, very, very probable. They are so probable, in fact, that we literally bet our lives on them on a daily basis, although perhaps unknowingly. Still, they are never certain, because as the case will always be, the set of evidences supporting them is finite, is obtained from a limited portion of the universe and is gleaned over a limited period of time. In other words, any claim held up by evidence is always merely provisional. For example, before the formulation of the Theory of Relativity and Quantum Theory, Classical Newtonian Mechanics was considered true. However, its truth was merely provisional, and when evidence arose that new theories were needed, and most especially when the required theories were discovered, Newtonian Mechanics had to go. In other words, when we say that the truth of a claim is provisional, what we mean is that the claim is true only as far as the present body of observation is concerned. Since the present body of observation is a finite set – and a finite set it will always be – certainty can never be had in science, hence the scientist’s skepticism and humility.

Full proof: an odd example

Let us now go to the nature of proof. As with evidence, an example would best explain the concept. Take the proposition “No odd number is divisible by two,” taking note that we are saying it in the context of standard arithmetic. For the sake of convenience, let us call the said proposition p. How do we know p is true? Well, 3 is an odd number and it is not divisible by two. In fact, 3 is a prime number and so it is divisible only by itself and unity. 5 and 7 are similarly odd and prime, so they’re not divisible by 2 as well. How about 9? Nope, 9 is similarly not divisible by 2, although it is divisible by 3. The same goes for 11, 13 and 15 – these bummers are not divisible by 2 either. But how about 17? Or 323,941? Or -5? There are an infinite number of odd integers, and there’s no point testing each of them if they are or are not divisible by 2. After all, proposition p claims that all of the infinitely many odd numbers are not divisible by 2. One could say that it is in the very definition of odd numbers that they are not divisible by 2, but that’s either begging the question or rigging it. A better solution to the problem could be achieved by way of a proof.

To proceed with the proof, we first note that odd numbers are integers of the form 1+2n, where n is any integer. In other words, odd numbers are what you get when you add twice an integer to the number one – odd numbers are every other integer with respect to the number one. 1+2=3 is an odd number, and so is 1-8 =-7. We now proceed with the proof:

1. When an integer a is divisible by an integer b, then a÷b is an integer;

2. Let x=1+2n, where n is any integer;

3. All odd numbers are of the form x=1+2n;

4. If x is divisible by 2, then x÷2 is an integer;

5. From (2) and (4), it follows that x÷2=(1+2n)÷2;

6. Using the distributive property of division over addition, it follows that (1+2n)÷2 = (1÷2) + (2n÷2);

7. Using the rule of cancellation, we know that 2n÷2=n;

8. Using the definition of fractions, we know that 1÷2=½;

9. In view of (7) and (8), it follows that (1+2n)÷2=½ + n;

10. In view of (5), we therefore have x÷2 = ½ + n;

11. Given (2) and (3), it follows that x is an odd number;

12. Since n is any integer, and the sum of an integer and a proper fraction is never an integer, it follows that x÷2 is not an integer;

Therefore: Whenever a number is odd, then it is not divisible by 2.

Given the above example, the following things can be said about a proof. First, a proof is a system of logically linked propositions that show that a conclusion follows necessarily from a given set of premises. In the example given, we have shown by proof that proposition p is necessarily true given that we are working within the context of standard arithmetic – you cannot believe in arithmetic and not believe the truth of p. Second, a million “evidences” for p will not be enough to establish its truth – a million odd numbers indivisible by 2 is not enough to show that p is true. However, a single valid proof, such as the one given above, is enough and is exactly what is needed. Third, not only is the truth of p necessary if we accept arithmetic, the proof above, if it is valid, establishes the truth of p as being absolute, certain and final – the truth of The Law of Pebble Addition can never acquire such attributes. Many philosophers would describe p as being true “in every conceivable universe”. If imaginary universes are not your type, think of the proof above as establishing the truth of p a priori, meaning, without reference to experience. Finally, and perhaps most importantly, proofs are only meaningful within the context of a formal system. A formal system (call it an axiomatic system if you want to impress your audience) is a system of propositions (theorems, lemmas, corollaries) that are logically deduced from the set of axioms and definitions of the system. Arithmetic is an example of a formal system, and the proposition p is an example of a theorem in arithmetic. Among other things, the formal system of arithmetic contains the definition of integers and odd numbers, the definition of addition, the distributive property of division over addition and the many axioms of addition. The proof given above will be meaningless if not taken within the context of arithmetic.

Catching for one’s breath: a brief recap

Once again, the evidences for a claim increase the probability of its truth while the proof of a claim shows that the claim follows necessarily from the premises. Additionally, the proof of a particular claim makes its truth final while the evidences supporting a claim lend the claim a provisional status. The type of reasoning used in making a formal proof is called deduction, while the type of reasoning used in inferring a conclusion from a body of evidences is called induction. Also, one important thing to remember about proofs is that they are meaningful only within the context of a specified formal system. Examples of formals systems are arithmetic, Euclidian geometry, analytical mechanics, calculus, Boolean logic, Java (the programming language, not the coffee) and chess. In the example we used above, the proof that p is true was made within the formal system of standard arithmetic (that is, using the axioms, definitions, properties and proven theorems of arithmetic). When the formal system is different, the proof of p presented above will no longer be valid or even meaningful.

Full proof 2: association is the key

To be able to compare and contrast proof and evidence much better, consider the following statement in arithmetic: “2+3=5”. Let us call this equation q. How do we know that q is true? As with the previous example, a proof is what we need.

1. 2=1+1;

2. 3=1+1+1;

3. 5=1+1+1+1+1;

4. In view of (1) and (2), we write 2+3=(1+1)+(1+1+1);

5. However, addition is associative, so (1+1)+(1+1+1) = 1+1+1+1+1;

6. In view of (4), we conclude that 2+3=1+1+1+1+1;

7. In view of (3) and (6), we conclude that 5=2+3;

Therefore: Using the symmetric property of equality, we finally conclude that 2+3=5.

We have therefore proven q to be true within the context of arithmetic. And because we have used a proof to show q as true, we can then say that q is true certainly, absolutely, eternally and a priori.

Now, compare equation q (“2+3=5”) with the Law of Pebble Addition (“2 pebbles plus 3 pebbles always and everywhere equals 5 pebbles”). Although they look akin to each other, the difference between them is very great. Recall that the Law of Pebble Addition (LPA) is a generalization that is inductively inferred, is merely probable and is provisional. On the other hand, equation q is logically deduced, is absolutely certain and is eternally true. Also, the LPA is supported by empirical evidence (that is, by repeated experience) while equation q is held up by a proof that is valid a priori (that is, without reference to experience).

However – and this is a very important however – the LPA is a very good generalization. Also, the generalization can be extended to include not only pebbles but many other things as well – we can formulate a Law of Peanut Addition (provided no one eats the peanuts), or a Law of Fundie Addition (provided the fundies don’t burn each other up on account of heresy) or other addition laws for material objects. We can also expand the generalization so that it will read “x objects plus y objects always and everywhere equals z objects”, granted x+y=z. We can do this because we have evidence to support the supposition that arithmetic addition can be used to model many kinds of material addition. In short, we have very strong reason to believe that pebbles (among many other things) add like integers. This belief, however, is merely provisional. That is, when a day comes when pebbles do not add like integers any more, it does not mean that there is something wrong with arithmetic or with the universe, it just means that pebbles simply do not add like integers any more.

As a matter of fact, many kinds of material addition cannot be modeled by arithmetic addition even today. The addition of moles of a substance is a splendid example. We know from high school chemistry that 1 mole of chloride (Cl-) and 1 mole of sodium ions (Na+) do not add to give 2 moles of table salt (NaCl). Rather, 1 mole of chloride and 1 mole of sodium ions give only 1 mole of NaCl. But is not 1+1=2? Well, yes, 1+1=2 in arithmetic. However, it appears that moles of a substance, unlike pebbles or apples, do not add like integers, so we cannot use arithmetic addition to model the physical operation of addition moles of a substance.

Also, take note that no observation can affect the truth of “1+1=2” or of equation q – these statements are true for all eternity (within the context of arithmetic, of course). Even if, and this is a big if, pebbles do not add like integers (for example, if 2 pebbles plus 3 pebbles gives 6 pebbles instead of 5), equation q (2+3=5) will still be true, and certainly and eternally so. Symmetrically, no empirical generalization can ever inherit the certain and eternal verity of arithmetic; even if “2+3=5” is certainly and eternally true, it does not follow that the LPA (“2 pebbles plus 3 pebbles equals 5 pebbles”) is eternally true. The LPA will always be provisional and merely probable (albeit very, very probable) because it will always be held up by a finite and limited body of evidences.

We must count ourselves fortunate that pebbles, apples and oranges add like integers. It really could have been otherwise, since the relationship between material addition and arithmetic addition is not one of necessity.

Truth and consequence

In colloquial language, we often use ‘proof’ and ‘evidence’ interchangeably, as in the conversation below.

A: I believe that the world is more than six thousand years old.

B: You’ve got proof?

A: I’ve got lots of proof. There’s radioactive dating, for one. And there are all sorts of archeological and paleontological proofs, too.

Also consider the statement, “I know God is real, and I have direct proof: I have a personal relationship with him.” And we often hear people speaking about the “proof of evolution” or “the proofs for the existence of God” or “the proofs that my boyfriend/girlfriend loves me”.

Colloquially, all of these uses are acceptable. However, many problems and controversies in philosophy and science, such as the existence of God, can discussed intelligently only using a more carefully structured language. Such a language must be sensitive to the fact that the truth of claims can be established either inductively or deductively. That is, a truth claim can either be supported by evidence or, if it is part of a formal system, it can be proved within the context of that formal system. Much is lost if we lose the distinction between proving a proposition and providing evidence for it. Since the goal of argument and philosophizing is truth – we do not argue or philosophize simply to stroke our ego, to masturbate intellectually – it would behoove us all to adopt a more structured vocabulary.

Now, if we are careful with our use of the words ‘proof’ and ‘evidence’, we find that calling St. Thomas’ Cosmological “Proofs” for the existence of God as proofs is to speak nonsense. The same goes for the Argument from Design, its kin the Fine-Tuned Universe Argument, the Argument from Miracles and the Argument from Religious Experiences– none of these are proofs for the existence of God since none of them are within the context of a specified formal system, and none of them use deductive reasoning. The Cosmological Arguments (the so-called “Five Proofs” of St. Thomas Aquinas), and all the rest, are all bodies of evidences which supposedly support the claim that God exists. This means that if they hold up to scrutiny, then they merely establish God’s existence in a provisional manner. Additionally, the conclusions of these arguments can never be certain because if they are successful, they merely establish the high probability of the existence of God. This means treating the existence of God as a scientific and empirical problem is already the death of religion; simply seeking for evidence that God exists already implies that you can never be certain of the existence of God.

Only two classes of arguments for the existence of God come near to being proofs, and these are the Ontological Proofs and the Transcendental Proofs for the existence of God. The Ontological Proofs and the Transcendental Proofs are arguments that purport to show that God necessarily exists using deductive reasoning and a priori premises. If any of them are valid, then they supposedly show that one cannot be rational and at the same time deny the existence of God. One fatal problem with these arguments is that the formal systems within which they are to be understood are not specified. This means that all their manipulations of ideas and symbols can be easily assailed. Notice that many steps in the proof for p and q cannot be understood if they are not taken in the context of arithmetic. Since arithmetic is a formal system, all symbolic operations, such as addition and its many properties, are well defined. The symbols themselves are rigidly defined and there is no room for ambiguity. The same cannot be said of the operations, symbols and symbolic manipulations of the Ontological and Transcendental Proofs, which is why I said that any step in these arguments can easily be assailed.

Another important consequence of the distinction between proofs and evidence is that science never proves anything; proofs are reserved for mathematicians, logicians and philosophers. I heard theologians were once part of the proving gang, but apparently they’ve been debarred since the Enlightenment. Now, this follows that all the conclusions of science are merely probable, provisional and pragmatically true. We accept the law of conservation of energy, for example, not because we have proven it and are thus certain of it, but because we have enormous evidence for it, and because we go a long way by believing that it is true. The same must be said of the theory of evolution, quantum theory, general relativity, or the theory that your lover is the best lover in the whole universe.

What’s in a name?

“What’s in a name?

That which we call a rose

By any other name would smell as sweet”

-Juliet, Shakespeare’s Romeo and Juliet

Along the same lines, one might ask, “But what’s the point of bothering with these trivial differences? Indeed, what’s in a name? Will not a proof by any other name be as rigorous? And a body of evidence by any other name be as supportive?”

Well, first of all, the difference isn’t trivial. It can be subtle sometimes. The examples I gave above are the simple ones. And second, it’s actually important. It’s not mere word game, and it certainly isn’t mere sophistry or pedantry. Many discourses we enter in our daily lives involve the assessment of a proof or the judgment of a body of evidences. Knowing which is which, and applying this knowledge to your daily life, is certainly not a waste of time.

As I have said before, much is lost if we lose sight of the distinction between proof and evidence. So much, in fact, that a lot of debate and discourse regarding the existence of God or the truth of a particular scientific theory will be fruitless, if not nonsense, if we fail to take note of the important distinction.

A rose by any other name would indeed smell as sweet. But call the rose the uwakuka if that sounds fine to you. However, don’t expect people to understand you whenever you proclaim the beauty of the sweet uwakuka. We have the right to invent our very own private language, but it would be to our benefit to use the languages that exist around us, and to use them correctly. After all, what’s at stake is nothing less than the truth.

* * *

It is my strong belief that philosophical truth can be attained only through a discourse between intelligent but independent minds. Since such discourse will be at peril if we will not be uncompromising in our linguistic standards, I would argue that the careful use of language is one of the essential ingredients in a fruitful dialogue.

Let the fruitful dialogue continue.

References:

[1] Roger Scruton, Modern Philosophy; An Introduction and Survey, Pimlico, 2004.

[2] Howard Kahane, Logic and Contemporary Rhetoric, Wadsworth, 1984.

[3] Arthur Danto and Sidney Morgensberger (editors), Philosophy of Science, Meridian, 1964.

[4] Morton White (editor), The Age of Analysis, Mentor, 1955.

Sometimes – actually oftentimes – we can get pretty sloppy and careless in our use of words.

Take the use of the words “proof” and “evidence”. Proof and evidence, like speed and velocity, or theory and guess, have colloquial definitions that often lead to confusion. In order to smooth the progress of communication and avoid misunderstanding, these words have been given technical definitions in science and philosophy. For example, speed is defined as the magnitude of velocity; the latter is a vector, the former is the scalar magnitude of that vector. Also, a scientific theory is not simply a guess; rather, it is a system of ideas constructed from a verified set of generalizations and observations. In the same way, scientists and philosophers use the words proof and evidence to designate two very different things. For example, we prove a mathematical theorem instead of “finding evidences” for its truth, while we accumulate the evidence for a particular scientific theory but we never “prove” a theory.

What’s the difference? The distinction is best illustrated by examples.

The calculus of pebbles: a rocky example

The fist example illustrates the nature of scientific evidence. Suppose you are walking on a beach when you decide to pick up exactly two pebbles and put them in your pocket. You then walk a few paces forward before picking up three more pebbles and putting them in the same pocket you put the first two pebbles. Again, you walk several paces forward before you decide to sit on a boulder by the shore. You fish for the pebbles in your pocket and count how many there are, and you find that there are five pebbles all in all. Since you remember that there were initially no pebbles inside your pocket before you picked up the first two, you are led to the hypothesis that two pebbles plus three pebbles will give you five pebbles. You throw the five pebbles and go home. The next day you decide to perform the same experiment: you walk along the same shore as the day before; you pick up two pebbles and put them in your initially empty pocket; you walk a little bit; you pick up three pebbles and put them in your pocket; you then walk a little bit, sit down on the boulder and finally count how many pebbles you end up with. Again, you find that you end up with five pebbles. You do this thing every day for a whole week, even a month, and each time you performed the same damned experiment, the result always confirms your hypothesis that two pebbles plus three pebbles equals five pebbles. You then decide to make the following scientific generalization based on your findings: “Two pebbles plus three pebbles always equals five pebbles.”

This generalization was inspired by the evidence for its truth. But note the ‘always’ that appears in the generalization. This ‘always’ is what makes the generalization risky, and as a consequence interesting and useful. It is interesting because it tells something about the workings of the universe (about the way pebbles add in our universe, for example). And it is useful because whenever an observation seems to contradict this generalization, an interesting problem arises, problems such as: “Why did I end up with one less pebble? Perhaps there’s a hole in my pocket.” You check your pocket and lo and behold, you find the hole that explains the missing pebble.

Notice that the more confirmations of the generalization you get, the more evidence you have of its truth. Notice also that the convincingness of the evidence is increased when it is obtained by someone else other than you and when the beach from which the evidence is obtained is a very different kind of beach from the beach that first inspired the generalization. Using these much more convincing evidence one can then formulate the broader generalization: “Two pebbles plus three pebbles always and everywhere gives five pebbles.” This last generalization is obviously riskier, but is also more interesting and more useful than the former.

Now, after perhaps a thousand such experiments, you get sick and tired of repeating the same wretched experiment that gives the same results over and over again that you say to yourself, “Well, since my generalization seems to be valid always and everywhere, let me give it a more imposing name, say ‘The Law of Pebble Addition’, and let me leave it at that. I have more important things to do with my life than counting pebbles.”

A very wise decision; I would recommend this course of action to anyone. However, the following questions can be asked with justice: How certain are you of the truth of your Law of Pebble Addition? How can you be absolutely sure of its validity, generality and universality? Does it apply today in the Andromeda Galaxy, or in the farthest part of the universe where pebbles are found? Did it apply back in the Precambrian Period, or will it apply five billion years into the future?

If you are honest and careful enough, your answer would be something like this: “No, I am not certain of its truth, because although I have a lot of evidence of it being true, the body evidence is nonetheless finite. I have never been to the Andromeda Galaxy, let alone the farthest part of the universe where pebbles are found. Also, I was not alive during the Precambrian, and I will not be alive five billion years into the future. Consequently, I cannot be certain about the generality and universality of the Law of Pebble Addition.” Of course you should add, “But I have a mountain of evidence that I can marshal to support my Law. And if you want, you can test it in places where I haven’t, perhaps using procedures I haven’t used. That way, you can convince yourself that the Law holds good always and everywhere, although I don’t recommend it to you, since I’ve already done the experiments and I think it’s a waste of your time if you repeat it. But that’s your call.”

We see from the example given above that evidences are observations that support a particular hypothesis. An empirical generalization or hypothesis, such as the Law of Pebble Addition, is inferred from the given body of evidence using the process of induction. Induction or inductive reasoning works in such a way that the more evidence you can gather to support a particular hypothesis, the greater becomes the probability of that hypothesis being true, which means the more you can bet on it. Some empirical generalizations, such as the conservation of energy, the conservation of angular momentum and the principle of invariance, are probable to the degree that they are often times called “Laws of Nature” (take note of the grave capital letters). Calling them Laws of Nature simply means that they are very, very, very probable. They are so probable, in fact, that we literally bet our lives on them on a daily basis, although perhaps unknowingly. Still, they are never certain, because as the case will always be, the set of evidences supporting them is finite, is obtained from a limited portion of the universe and is gleaned over a limited period of time. In other words, any claim held up by evidence is always merely provisional. For example, before the formulation of the Theory of Relativity and Quantum Theory, Classical Newtonian Mechanics was considered true. However, its truth was merely provisional, and when evidence arose that new theories were needed, and most especially when the required theories were discovered, Newtonian Mechanics had to go. In other words, when we say that the truth of a claim is provisional, what we mean is that the claim is true only as far as the present body of observation is concerned. Since the present body of observation is a finite set – and a finite set it will always be – certainty can never be had in science, hence the scientist’s skepticism and humility.

Full proof: an odd example

Let us now go to the nature of proof. As with evidence, an example would best explain the concept. Take the proposition “No odd number is divisible by two,” taking note that we are saying it in the context of standard arithmetic. For the sake of convenience, let us call the said proposition p. How do we know p is true? Well, 3 is an odd number and it is not divisible by two. In fact, 3 is a prime number and so it is divisible only by itself and unity. 5 and 7 are similarly odd and prime, so they’re not divisible by 2 as well. How about 9? Nope, 9 is similarly not divisible by 2, although it is divisible by 3. The same goes for 11, 13 and 15 – these bummers are not divisible by 2 either. But how about 17? Or 323,941? Or -5? There are an infinite number of odd integers, and there’s no point testing each of them if they are or are not divisible by 2. After all, proposition p claims that all of the infinitely many odd numbers are not divisible by 2. One could say that it is in the very definition of odd numbers that they are not divisible by 2, but that’s either begging the question or rigging it. A better solution to the problem could be achieved by way of a proof.

To proceed with the proof, we first note that odd numbers are integers of the form 1+2n, where n is any integer. In other words, odd numbers are what you get when you add twice an integer to the number one – odd numbers are every other integer with respect to the number one. 1+2=3 is an odd number, and so is 1-8 =-7. We now proceed with the proof:

1. When an integer a is divisible by an integer b, then a÷b is an integer;

2. Let x=1+2n, where n is any integer;

3. All odd numbers are of the form x=1+2n;

4. If x is divisible by 2, then x÷2 is an integer;

5. From (2) and (4), it follows that x÷2=(1+2n)÷2;

6. Using the distributive property of division over addition, it follows that (1+2n)÷2 = (1÷2) + (2n÷2);

7. Using the rule of cancellation, we know that 2n÷2=n;

8. Using the definition of fractions, we know that 1÷2=½;

9. In view of (7) and (8), it follows that (1+2n)÷2=½ + n;

10. In view of (5), we therefore have x÷2 = ½ + n;

11. Given (2) and (3), it follows that x is an odd number;

12. Since n is any integer, and the sum of an integer and a proper fraction is never an integer, it follows that x÷2 is not an integer;

Therefore: Whenever a number is odd, then it is not divisible by 2.

Given the above example, the following things can be said about a proof. First, a proof is a system of logically linked propositions that show that a conclusion follows necessarily from a given set of premises. In the example given, we have shown by proof that proposition p is necessarily true given that we are working within the context of standard arithmetic – you cannot believe in arithmetic and not believe the truth of p. Second, a million “evidences” for p will not be enough to establish its truth – a million odd numbers indivisible by 2 is not enough to show that p is true. However, a single valid proof, such as the one given above, is enough and is exactly what is needed. Third, not only is the truth of p necessary if we accept arithmetic, the proof above, if it is valid, establishes the truth of p as being absolute, certain and final – the truth of The Law of Pebble Addition can never acquire such attributes. Many philosophers would describe p as being true “in every conceivable universe”. If imaginary universes are not your type, think of the proof above as establishing the truth of p a priori, meaning, without reference to experience. Finally, and perhaps most importantly, proofs are only meaningful within the context of a formal system. A formal system (call it an axiomatic system if you want to impress your audience) is a system of propositions (theorems, lemmas, corollaries) that are logically deduced from the set of axioms and definitions of the system. Arithmetic is an example of a formal system, and the proposition p is an example of a theorem in arithmetic. Among other things, the formal system of arithmetic contains the definition of integers and odd numbers, the definition of addition, the distributive property of division over addition and the many axioms of addition. The proof given above will be meaningless if not taken within the context of arithmetic.

Catching for one’s breath: a brief recap

Once again, the evidences for a claim increase the probability of its truth while the proof of a claim shows that the claim follows necessarily from the premises. Additionally, the proof of a particular claim makes its truth final while the evidences supporting a claim lend the claim a provisional status. The type of reasoning used in making a formal proof is called deduction, while the type of reasoning used in inferring a conclusion from a body of evidences is called induction. Also, one important thing to remember about proofs is that they are meaningful only within the context of a specified formal system. Examples of formals systems are arithmetic, Euclidian geometry, analytical mechanics, calculus, Boolean logic, Java (the programming language, not the coffee) and chess. In the example we used above, the proof that p is true was made within the formal system of standard arithmetic (that is, using the axioms, definitions, properties and proven theorems of arithmetic). When the formal system is different, the proof of p presented above will no longer be valid or even meaningful.

Full proof 2: association is the key

To be able to compare and contrast proof and evidence much better, consider the following statement in arithmetic: “2+3=5”. Let us call this equation q. How do we know that q is true? As with the previous example, a proof is what we need.

1. 2=1+1;

2. 3=1+1+1;

3. 5=1+1+1+1+1;

4. In view of (1) and (2), we write 2+3=(1+1)+(1+1+1);

5. However, addition is associative, so (1+1)+(1+1+1) = 1+1+1+1+1;

6. In view of (4), we conclude that 2+3=1+1+1+1+1;

7. In view of (3) and (6), we conclude that 5=2+3;

Therefore: Using the symmetric property of equality, we finally conclude that 2+3=5.

We have therefore proven q to be true within the context of arithmetic. And because we have used a proof to show q as true, we can then say that q is true certainly, absolutely, eternally and a priori.

Now, compare equation q (“2+3=5”) with the Law of Pebble Addition (“2 pebbles plus 3 pebbles always and everywhere equals 5 pebbles”). Although they look akin to each other, the difference between them is very great. Recall that the Law of Pebble Addition (LPA) is a generalization that is inductively inferred, is merely probable and is provisional. On the other hand, equation q is logically deduced, is absolutely certain and is eternally true. Also, the LPA is supported by empirical evidence (that is, by repeated experience) while equation q is held up by a proof that is valid a priori (that is, without reference to experience).

However – and this is a very important however – the LPA is a very good generalization. Also, the generalization can be extended to include not only pebbles but many other things as well – we can formulate a Law of Peanut Addition (provided no one eats the peanuts), or a Law of Fundie Addition (provided the fundies don’t burn each other up on account of heresy) or other addition laws for material objects. We can also expand the generalization so that it will read “x objects plus y objects always and everywhere equals z objects”, granted x+y=z. We can do this because we have evidence to support the supposition that arithmetic addition can be used to model many kinds of material addition. In short, we have very strong reason to believe that pebbles (among many other things) add like integers. This belief, however, is merely provisional. That is, when a day comes when pebbles do not add like integers any more, it does not mean that there is something wrong with arithmetic or with the universe, it just means that pebbles simply do not add like integers any more.

As a matter of fact, many kinds of material addition cannot be modeled by arithmetic addition even today. The addition of moles of a substance is a splendid example. We know from high school chemistry that 1 mole of chloride (Cl-) and 1 mole of sodium ions (Na+) do not add to give 2 moles of table salt (NaCl). Rather, 1 mole of chloride and 1 mole of sodium ions give only 1 mole of NaCl. But is not 1+1=2? Well, yes, 1+1=2 in arithmetic. However, it appears that moles of a substance, unlike pebbles or apples, do not add like integers, so we cannot use arithmetic addition to model the physical operation of addition moles of a substance.

Also, take note that no observation can affect the truth of “1+1=2” or of equation q – these statements are true for all eternity (within the context of arithmetic, of course). Even if, and this is a big if, pebbles do not add like integers (for example, if 2 pebbles plus 3 pebbles gives 6 pebbles instead of 5), equation q (2+3=5) will still be true, and certainly and eternally so. Symmetrically, no empirical generalization can ever inherit the certain and eternal verity of arithmetic; even if “2+3=5” is certainly and eternally true, it does not follow that the LPA (“2 pebbles plus 3 pebbles equals 5 pebbles”) is eternally true. The LPA will always be provisional and merely probable (albeit very, very probable) because it will always be held up by a finite and limited body of evidences.

We must count ourselves fortunate that pebbles, apples and oranges add like integers. It really could have been otherwise, since the relationship between material addition and arithmetic addition is not one of necessity.

Truth and consequence

In colloquial language, we often use ‘proof’ and ‘evidence’ interchangeably, as in the conversation below.

A: I believe that the world is more than six thousand years old.

B: You’ve got proof?

A: I’ve got lots of proof. There’s radioactive dating, for one. And there are all sorts of archeological and paleontological proofs, too.

Also consider the statement, “I know God is real, and I have direct proof: I have a personal relationship with him.” And we often hear people speaking about the “proof of evolution” or “the proofs for the existence of God” or “the proofs that my boyfriend/girlfriend loves me”.

Colloquially, all of these uses are acceptable. However, many problems and controversies in philosophy and science, such as the existence of God, can discussed intelligently only using a more carefully structured language. Such a language must be sensitive to the fact that the truth of claims can be established either inductively or deductively. That is, a truth claim can either be supported by evidence or, if it is part of a formal system, it can be proved within the context of that formal system. Much is lost if we lose the distinction between proving a proposition and providing evidence for it. Since the goal of argument and philosophizing is truth – we do not argue or philosophize simply to stroke our ego, to masturbate intellectually – it would behoove us all to adopt a more structured vocabulary.

Now, if we are careful with our use of the words ‘proof’ and ‘evidence’, we find that calling St. Thomas’ Cosmological “Proofs” for the existence of God as proofs is to speak nonsense. The same goes for the Argument from Design, its kin the Fine-Tuned Universe Argument, the Argument from Miracles and the Argument from Religious Experiences– none of these are proofs for the existence of God since none of them are within the context of a specified formal system, and none of them use deductive reasoning. The Cosmological Arguments (the so-called “Five Proofs” of St. Thomas Aquinas), and all the rest, are all bodies of evidences which supposedly support the claim that God exists. This means that if they hold up to scrutiny, then they merely establish God’s existence in a provisional manner. Additionally, the conclusions of these arguments can never be certain because if they are successful, they merely establish the high probability of the existence of God. This means treating the existence of God as a scientific and empirical problem is already the death of religion; simply seeking for evidence that God exists already implies that you can never be certain of the existence of God.

Only two classes of arguments for the existence of God come near to being proofs, and these are the Ontological Proofs and the Transcendental Proofs for the existence of God. The Ontological Proofs and the Transcendental Proofs are arguments that purport to show that God necessarily exists using deductive reasoning and a priori premises. If any of them are valid, then they supposedly show that one cannot be rational and at the same time deny the existence of God. One fatal problem with these arguments is that the formal systems within which they are to be understood are not specified. This means that all their manipulations of ideas and symbols can be easily assailed. Notice that many steps in the proof for p and q cannot be understood if they are not taken in the context of arithmetic. Since arithmetic is a formal system, all symbolic operations, such as addition and its many properties, are well defined. The symbols themselves are rigidly defined and there is no room for ambiguity. The same cannot be said of the operations, symbols and symbolic manipulations of the Ontological and Transcendental Proofs, which is why I said that any step in these arguments can easily be assailed.

Another important consequence of the distinction between proofs and evidence is that science never proves anything; proofs are reserved for mathematicians, logicians and philosophers. I heard theologians were once part of the proving gang, but apparently they’ve been debarred since the Enlightenment. Now, this follows that all the conclusions of science are merely probable, provisional and pragmatically true. We accept the law of conservation of energy, for example, not because we have proven it and are thus certain of it, but because we have enormous evidence for it, and because we go a long way by believing that it is true. The same must be said of the theory of evolution, quantum theory, general relativity, or the theory that your lover is the best lover in the whole universe.

What’s in a name?

“What’s in a name?

That which we call a rose

By any other name would smell as sweet”

-Juliet, Shakespeare’s Romeo and Juliet

Along the same lines, one might ask, “But what’s the point of bothering with these trivial differences? Indeed, what’s in a name? Will not a proof by any other name be as rigorous? And a body of evidence by any other name be as supportive?”

Well, first of all, the difference isn’t trivial. It can be subtle sometimes. The examples I gave above are the simple ones. And second, it’s actually important. It’s not mere word game, and it certainly isn’t mere sophistry or pedantry. Many discourses we enter in our daily lives involve the assessment of a proof or the judgment of a body of evidences. Knowing which is which, and applying this knowledge to your daily life, is certainly not a waste of time.

As I have said before, much is lost if we lose sight of the distinction between proof and evidence. So much, in fact, that a lot of debate and discourse regarding the existence of God or the truth of a particular scientific theory will be fruitless, if not nonsense, if we fail to take note of the important distinction.

A rose by any other name would indeed smell as sweet. But call the rose the uwakuka if that sounds fine to you. However, don’t expect people to understand you whenever you proclaim the beauty of the sweet uwakuka. We have the right to invent our very own private language, but it would be to our benefit to use the languages that exist around us, and to use them correctly. After all, what’s at stake is nothing less than the truth.

* * *

It is my strong belief that philosophical truth can be attained only through a discourse between intelligent but independent minds. Since such discourse will be at peril if we will not be uncompromising in our linguistic standards, I would argue that the careful use of language is one of the essential ingredients in a fruitful dialogue.

Let the fruitful dialogue continue.

References:

[1] Roger Scruton, Modern Philosophy; An Introduction and Survey, Pimlico, 2004.

[2] Howard Kahane, Logic and Contemporary Rhetoric, Wadsworth, 1984.

[3] Arthur Danto and Sidney Morgensberger (editors), Philosophy of Science, Meridian, 1964.

[4] Morton White (editor), The Age of Analysis, Mentor, 1955.

## 2 Commments:

teach:

The LPA "two pebbles plus three pebbles gives five pebbles is valid always and everywhere" can be demonstrated to be false, and how and where and when a universal generalization can not be made - as it bumps into an exception.

1) In a galaxy and planetary system where pebbles can reproduce asexually much like bacteria. (They can be slightly bigger. They can be rocks.) You count them again and voila they've multiplied.

2) In a galaxy and planetary system where pebbles have sexual reproductive systems, that can fertilize impregnate gestate and give rise to new pebbles. You count them again and voila they've increased!

@Carl

You said, "The LPA 'two pebbles plus three pebbles gives five pebbles is valid always and everywhere' can be demonstrated to be false, and how and where and when a universal generalization can not be made - as it bumps into an exception." Kindly make your meaning a little clearer.

However, if you mean that the LPA may always bump into an exception some time in the future, then I entirely agree with you. In fact, that's one of the main points of the essay.

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