Mathematical Truth and Philosophy

Curtis J. Metcalfe
April 24, 2009

There lies at the heart of mathematics an assumed premise that it provides truth. We are taught
that the truth of mathematics is real, absolute, and indubitable. Mathematics, to the dismay of so many students, has right and wrong answers. It has explicit values and formulas. It is true and works at all times and for all people. It is eternal. Isnʼt it?

Math homework has been handed back covered in red ink that proclaims, “Yes!” But how can we
know that the truths of mathematics are, in fact, true? And even if they are, how do we have access to them? And do they give us truth about the world outside of mathematics, or is it merely a closed system, true but vacuous? By exploring first logic and the use of language, as well as the history of philosophy of mathematics, we can move toward the understanding of more complex systems of symbols and probabilities and can see how, in logic and philosophy, we attempt to derive truth from our language in much the same way we derive truth from mathematics and, indeed, vice versa.

RULES OF LOGIC
To the uninitiated it may seem strange that the most basic element of reason is actually an
argument. But in this way we are not arguing (in the usual sense) so much as arguing for a
conclusion.1 Properly understood, an argument is a defense of a particular position where we use one or more premise(s) to justify our arrival at a conclusion. We can take the following argument for example:
1. All human beings are born, not hatched.
2. I am a human being.
3. Therefore, I was born, not hatched.
So then, (1) and (2) are the premises and (3) is the conclusion. We see that if both of the premises are correct, the conclusion must be correct. It is utterly unavoidable. However, we do not have quite the truth we are looking for yet, since arguments are only (in this sense) defined by their form, not their content. An argument with false premises and a false conclusion, in proper form, is still valid. In this way, an argument cannot be said to be true or false, only valid or invalid. Only statements can be true or false.

This is a fundamental truth of symbolic logic. So to get sentences into a form manageable by a
system of formal logic, we use variables and symbols, whereby a sentence that reads, “either I am awake or I am asleep,” can be represented by, “p or q,” where p represents “I am awake,” and q represents “I am asleep.” Or, written in symbolic form, p∨q, where ∨ is the connective “or.”
Further, if p is true and q is true, then p∨q is true; if either p or q is false, p ∨ q remains true; if
both p and q are false, p ∨ q is false. We will return to the use of symbols later on.

We can set up such universal scenarios as these with truth tables for many other argument forms
and rules of inference, but it is sufficient for our purposes to understand that we can, on at least the level shown so far, use declaratory statements in simple forms, universally replaced by variables and computed in a way that shows the validity of our argument. Nevertheless, we must mind our propositions and carefully examine how we set up our arguments. For example, the argument form if p, then q; and p, therefore q is obviously valid. It is also valid for us to say, if p, then q; and ~q, therefore ~p. However, to conclude either ~p therefore ~q or q therefore p are invalid.2 But, one might wonder, what have we really shown to be true? At most, we have come up with some statements that under some circumstance may be true and at other times may not be. These statements surely are not necessary but contingent on some other set of facts, which is not the way we generally take mathematics to work. Logic, though it can be written in a mathematical form (as we will see with more complexity later), is only superficially related to mathematics in most instances. Clearly we do not place mathematics and language in the same discipline, so there must be differences.

RULES OF MATHEMATICS
Humanist philosopher and mathematician Reuben Hersch calls the method of mathematics
“conjecture and proof.”3 In this sense it is traditionally only mathematics that can claim to have a “proof” of anything, where a mathematical statement is shown to be true, necessarily. We say that 2+2=4 necessarily, and anyone who understand the terms “2” “+” and “4” as (1+1)+(1+1)
cannot deny the fact. But explaining exactly what a proof is can be a difficult if not impossible task, outside of understanding that it is, at least, how one convinces another of a theorem or conclusion.4 So it is this type of proof we are hoping to be able to apply to other areas, where we can show our conclusions in such a way that they are unavoidable, beyond doubt.

HISTORY OF MATHEMATICS AND PHILOSOPHY
The history of mathematics and philosophy, (philosophy of mathematics) is a lesson far too complex to address here. However, a few thinkers in particular can be seen as points on our line worth stopping to consider. The giants in both ancient and modern philosophy have sought to incorporate mathematics into their ideas. Platoʼs idea of mathematical objects existing immaterially in the perfect world of the Forms where the ideal version of everything exists is largely the framework by which mathematicians still work, aware of it or not. The absolute truths of mathematics exist, thought Plato, in the same way the perfect form of every abstract idea does. When one thinks of the color blue, it isnʼt some certain blue, but the very form of blue. Likewise, the idea of table must exist perfectly and immaterially in the realm of the forms in order that we recognize every individual, different table as having those categories essential for tableness. Or, when one has the idea of tiger it is surely not just one individual tiger, so that an actual tiger would cease to be recognizable because of its deviation from that one tiger in our mind. Somehow, we must have a notion, Plato thought, of tigerness, and it must be in the immaterial world of the Forms. Other philosophers like Descartes attempted to arrive at things in life so certain and indubitable, so clear and distinct, as the truths of mathematics, and Spinoza modeled his Ethics in a Euclidian fashion of axioms and theorems. David Hume accepted the truths of math only insofar as they gave vacuous truth, mere tautologies. The truth of 7+5=12 may hold, but gives us no truth about the universe. Math is like a game that we invented with arbitrary rules, so while even if it is true, it is contingent.

We can quite easily, perhaps, feel like Plato might have had a critical idea, but the idea of an
immaterial world presents some problems. How do we interact with it? How do immaterial and
material coincide? Does it not seem just as plausible that we abstract the general idea from the
specific experience? How can we have an idea of a tiger with no specific qualities, yet still call it a
tiger? Even if every mathematical truth is existent somewhere, somehow, how could we ever know it? It seems like there are problems, to be sure. But is the alternative better? Is math only a
game? We could imagine this being so in light of the use of language that is so necessary in
mathematics, but we also realize that mathematics just simply works too well, too often, and
(especially in physics and cosmology) too far away. But let us, for now, consider more deeply how mathematics applies to truth and philosophy and place the former questions on hold.

APPLICATION
So although there is considerable debate about the nature of mathematics and the absoluteness
thereof, we may consider certain axioms and rules in our language as they are used in areas such
as symbolic logic and probability calculus. By using symbols in a system of sentential, deductive
logic, we see a system very reminiscent of algebra. Indeed, “mathematics uses deductive logic to
get its results.”5 What follows will be several examples of each, as well as the mathematic
principles that help make them so.

SYMBOLIC LOGIC
Sentential logic deals with translating sentences with a truth-value into symbols.6 We symbolize
words like and, or, then, not and therefore.7 We can use the variables p and q to represent generic sentences, and constants (A,B,C, etc.) to represent specific sentences. Consider the sentence, “Alfred and Bill are funny.” We could say it differently, that, “Alfred is funny and Bill is funny,” or (A⋅B). If we say “Alfred is funny and Bill is not funny,” (A⋅¬B). Or, “Alfred is not funny and Bill is not funny,” (¬A⋅¬B) or (¬A⋅B). So the sentence “Alfred is funny or Bill is funny” is symbolized as (A∨B) and “Either Alfred is funny or Bill is funny, but not both is {(A∨B)⋅¬(A⋅B)}. Here one will notice the use of brackets and parentheses just as in an equation in algebra, and for the same sort of purposes. For example, (6× 2)÷3 ≠ 6×(2÷3). In the same way, (A⋅ B)∨C≠ A⋅(B∨C). To see what this looks like in predicate logic (where a property is ascribed to some individual entity), we can use the following argument.8

1.A temporal world exists. Te
2.God is omniscient. Og
3.If a temporal world exists, then if God is omniscient, God knows tensed facts. Te⊃(Og⊃Kg)
4.If God is timeless, He does not know tensed facts. (Tg⊃¬Kg)
¬(¬Kg)
5.Therefore, God is not timeless.
∴¬Tg

By assuming the truth of the statements, we can test for the validity of the argument. So, for
example, in (3) we see that if Og, then Kg, and since we established the truth of Og in (2) and the
first half in (1), then (3) is valid. In (4) we have if Tg then not Kg, but since we established in (3)
the truth of Kg, we can deduce (5), not not Kg, and therefore (6) not Tg.
Notice we have not determined the truth or falsity of the argument, only the validity. The validity of an argument is shown “if and only if it is not possible for all of its premises to be true and its conclusion false. If all premises of a valid argument are true, then its conclusion must be true also.”9 The conclusion is inescapable. So even though the truth of each premise may be dubitable, as long as it follows this form and these rules and the conclusion follows logically, it is valid. We can show one more example, also from an argument in Dr. Craigʼs “Timelessness and
Omnitemporality” where t represents any time prior to creation and n some finite interval of time:

1.If the past in infinite, then at t God delayed creating until t+n. Ip⊃Dg
2.If at t God delayed creating until t+n, then He must have had a good reason for doing so.
Dg⊃Rg
3. Ip⊃Rg
4.If the past in infinite, God cannot have had a good reason for delaying at t creating until t+n.
Ip⊃¬Rg
5. (Ip⊃Rg⋅ Ip⊃¬Rg)
6.Therefore, if the past in infinite, God must have had a good reason for delaying at t and God
cannot have had a good reason for delaying at t. Ip⊃(Rg⋅¬Rg)
7.Therefore, the past is not infinite. ∴¬Ip

(3)is not found in the argument explicitly, but is established by Hypothetical Syllogism from (1) and (2), where, if Ip then Dg and if Dg then Rg therefore (3) if Ip then Rg. And similarly, (5) is the addition of (3) and (4). In this way we could say symbolic logic looks like algebra. But could we say it has anything to do with numbers, or is it actually only a superficial similarity? To see a system of logic and philosophy that deals with mathematics in a more numerical way, we turn to probability.

PROBABILITY CALCULUS
Properly understood, a probability is the likeliness of something happening or not. In mathematics and philosophy we use probability calculations to determine how sound a belief is or how likely some outcome or another is. A probability will have a value between 0 and 1, with 0 being impossible and 1 being certain. What probability calculations give us, then, is a quantity with which to asses truth. Perhaps this is what we are looking for. First, consider the probability that two dependent events will have a certain outcome. We can write it in the following way to say that the probability of B given A equals the probability of B and A over the probability of A.
Prob (B| A)=Prob(B∩A)/Prob (A)

There are some general rules to consider 10:
1. Restricted Conjunction Rule
Prob (A⋅B)=Prob (A)×Prob (B)

2. General Conjunction Rule
Prob (A⋅B)=Prob (A)×Prob (B| A)

3. Restricted Disjunction Rule
Prob (A∨B)=Prob (A)+Prob (B)

4. General Disjunction Rule
Prob (A∨B)=Prob (A)+Prob (B)−Prob (A⋅B)

5. Prob (A⋅¬A)=0

6. Prob (A∨¬A)=1

7. Prob (¬A)=1−Prob(A)

But a more complicated probability is the inverse probability of Bayesʼ Theorem, which, in itʼs
general form is

Prob (q| p)= Prob (q)×Prob (p|q) / [Prob (q1)×Prob (p|q1)+…+ Prob (qn)×Prob (p|qn)]

But what does all of this probability calculus amount to? What does it give us? In their argument
against the claim that the fine-tuning of the universe can be used as evidence for an intelligent
designer, Michael Ikeda and Bill Jefferys have given the following argument in the form of
probability 11:

Prob (F&L&¬N)=Prob (L| F&¬N)Prob (F|¬N)Prob(¬N)<<1

What this says is that the probability of the conditions in the universe being life-Friendly, and the universe existing and containing Life and that the universe is not governed solely by Naturalistic laws equals the probability of the universe containing Life given that the universe is life-Friendly and not governed solely by Naturalistic laws times the probability of the universe being life-Friendly given that the universe is not governed solely by Naturalistic laws times the probability that the universe is not governed solely by Naturalistic laws is all less than 1. That the sentence is complex and far from our task at hand is obvious, but what must also be true is that, given all the multiplication of the probabilities, for it to be less than 1, at least one of Prob (L| F&¬N), Prob (F|¬N), or Prob (¬N) is quite small.12 In other words, some of the values (such as L) have a known value (in this case, 1). The entire argument is in fact conditioned against that value. With this argument Ikeda and Jeffereys hope to show that in any case the Prob (¬N) is very small, or that Prob (F) is also small. Whether they are correct (or on which count) is of course debatable (we are dealing in probabilities here), but it is clear that they are using the inverse probability to show their position.

CONCLUSION
All of these equations and calculuses are examples of the way we can use the form and method of
mathematics in our use of language and philosophy. But what have we learned? Is there anything that can be set in stone? Is anything indubitable? Whether the truths of mathematics are true at all times for all people in all places, (another galaxy?) or whether they are contingently true given some constants such as the gravitational pull or the laws of motion or a certain set of axioms, they seem to at least give us the ability to establish truth if only in limited ways. But does the fact that 2+2 might have a different sum in another possible world qualify it for the scrap heap of subjective judgements? It certainly seems not. So what nearly every philosopher and mathematician has assumed can be held given the fact that we can have such interdisciplinary application of formula and equation. Perhaps, then, it is not the case that mathematical truths must be eternal, but do we have need of such certainty? The conversation tends to drift away from the mathematician on the possibilities of a dualism which does not seem to arise from material causes or the ontological status of abstract and mathematical
objects, even if it is a hidden assumption of the working mathematician. That the mathematician or empirical scientist does not like the conclusion does not, of course, render it untrue, but it may present difficulties that need to be addressed. And indeed, the incredibly specialized and complex mathematics behind these assumptions may be outside the comfort of most philosophers, but one can confidently proceed on both fronts having established that mathematics and logic work in tandem to give us some analytic tools necessary to deduce truths about the workings of the world around us.

FOOTNOTES
1
Alan Hausman, Howard Kahane, Paul Tidman, Logic and Philosophy, (Thomson Wadsworth)
2007, p. 1
2
Antony Flew, How to Think Straight, (Prometheus Books, Amherst, New York) 1998, pp. 35-36.
3
Reuben Hersch,What is Mathematics, Really? (Oxford University Press) 1997, p. 5
4
For a more in depth look at the problems with proof and certainty see Hersh, chapter 4, and for the nature of mathematics see chapter 1. His argument is that mathematics are essentially social
constructs and do not transcend their own limitations and that numbers in our language operate
both as verbs and nouns and cannot have the sort of reliability we normally assign to them in any metaphysical or transcendent sense. The sum of the interior angles of a triangle are only 180
degrees exactly on a perfect triangle, and only on a Euclidian plane. Outside of Euclidian Geometry such a claim becomes dubitable. “Euclidʼs Fifth” axiom, if it is false, changes the very claim about triangles and their interior angles. However, he does not buy David Humeʼs contention that mathematics is “only a game” for various reasons, also external to our discussion. For our study, we will work inside the axioms traditionally set up, leaving the arbitrariness thereof for other discussions.
5
Hausman, et al., p. 21
6
Ibid., p. 22
7
We might, for example, see the following symbols and connectors: and ⋅; or ∨; if/then ⊃; if and
only if ≡; not ¬; therefore ∴
8
William Lane Craig, “Timelessness and Omnitemporality” Philosophia Christi, Series 2, Vol. 2, No. 1, 2000, pp. 29-33. His arguments are found only in the original sentence form, not in symbols, which have been added to show how these symbols work in argument and form.
9
Hausman, et al., p. 17.
10
Ibid., pp. 400-401.
11
Michael Ikeda & Bill Jefferys, “The Anthropic Principle Does Not Support Supernaturalism”, http://quasar.as.utexas.edu/anthropic.html, 2006. The paper is quite long and uses many different probabilistic arguments, only one of which is shown here as a use of the calculus with respect to an argument from philosophy.
12
Each of these gives a different scenario, such that the traditional concept of a deity is undermined with respect to the Anthropic Argument. None of them are particularly important to our discussion, as we are only attempting to show how the calculus applies, bridging the gap between the disciplines.