So, the arithmetical proposition is a statement about certain characteristics of things (closely connected to the concept of counting). But it abstracts from or ignores other characteristics (such as the potential for biological, chemical or physical interactions). The logical proposition is clear and unambiguous, but trivial. The possibilities in real world applications are regularly much more nuanced and complex.
Example: Proof by contradiction?
As a final example, take the assertion that two propositions that contradict each other cannot both be true. A corollary is the idea that the proposition that the demonstration that a statement logically leads to a contradiction constitutes a “proof” that the statement is not true (or vice versa). Are these concepts empirical, that is, do they reflect things that are true (or untrue) about the physical world?
Clearly, the propositions seem to be integral to the notion of rational thought and logic and, probably, inherent in the human mental processes. For example, Professor Arkes states that “[t]he law of contradiction expresses a necessary truth, and all efforts to refute it will fall into the embarrassment of self-contradiction.” First Things, p.51. This “necessary truth,” like the similar concept of causality (discussed above), may be “necessary” in terms of the functioning of the human mind, but is it necessarily “true”?9 It may be that we are incapable of conceiving of two contradictory propositions both being true, but our inability to conceive does not control the actual physical relationships of the world.
In fact, the concept of inconsistency itself seems necessarily to implicate human theories or models. We might say that we have never observed in nature an entity that is simultaneously dead and alive, as evidence that nature does not tolerate inconsistent or contradictory states. But, that claim requires definitions of “dead” and “alive”. One could imagine plausible definitions that overlap, allowing the alleged inconsistency to exist in the external world (say a definition of death that turns on brain activity and a definition of life that is based upon cell reproduction). If the definitions are structured to be mutually exclusive, then no entity could satisfy both—by definition. Take a different type of example. We recognize that a person could simultaneously be both “short” and “tall”, since those characteristics are relative and we could have two different points of references.
I note that scientists often use inconsistencies as a form of proof. A typical “proof” using this type of argument would be as follows: Proposition A is either true or false. Suppose that I can deduce (using deduction) that if A is false, then both B and not-B are true. Since that conclusion presents a contradiction that cannot be accepted, I conclude that A is true. Actually, the logic tells us only that A is not false, since that is the assumption that led to the contradiction. The conclusion that A is, therefore, true depends upon the initial assertion or assumption that A is either true of false. If A could be something else, then the conclusion that A is true is not established.
The Uses and Usefulness of Mathematics
The impressive success of modern technology is a result of the inexplicable fact that the physical world can be simulated by computational models and that greater and greater accuracy has been achieved through more and more sophisticated mathematics. See Penrose, Shadows of the Mind, p.203. It is the “fruitfulness” of mathematics that is, perhaps, the most surprising fact. Mathematics has been used not only to capture the patterns of observed phenomena, but as a guide to the discovery of new theories and models that have then led to the discovery of empirical patterns and relationships that had not previously been noticed. See id., pp.416–7.
This capability may seem particularly surprising if one accepts what has been said so far about mathematics being just one or more deductive system in which the implications of the initial assumptions are all necessarily contained in those assumptions. In that sense, all mathematics is circular. There can be nothing new derived by manipulating the logical system, only different ways of expressing the same thing. So, how can that be such a powerful tool?
Where a set of relationships in the observed world has been defined (or hypothesized) logically as a deductive theory, it may be the case that the logical rules of the theory can be represented mathematically, using existing mathematics or even creating mathematics to fit the situation. The mathematical representation of the theory often provides very significant pragmatic benefits. Mathematics is based upon a system of notation using symbols. The use of symbols in place of words and phrases forces precision and clarity, as well as providing great economy and simplicity in the formulation of propositions or assertions. The simplicity itself is important. As Whitehead observed, “[b]y relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and in effect increases the mental power.” An Introduction to Mathematics, p.59.
Furthermore, the mathematical rules allow extensive manipulation by hand and, even for calculations, by machine (including the computer). Moreover, the process of breaking logical relationships down into many small steps makes it easier to avoid errors in constructing predictions and theories or to check for errors after the fact. These are very significant practical benefits attainable by setting forth theories in mathematical form.
However, it also appears that the use of the mathematics may even enable the development of concepts that have useful analogs for or provide surprising insights consistent with the non-mathematical theory.
Let us be more specific. Physicists Cox and Forshaw assert that “[e]quations are the most powerful tools available to physicists in their quest to understand nature.” Why Does E=mc2?, p.21. An equation sets out on each side of the “equals sign” two formulations that, when reduced to numerical values, will be equal in quantity. Sometimes the equality is “by definition,” in which case the equation can be considered to be a definition of one of the terms. In other cases, the equation expresses an empirical relationship.
It may be hard to tell the two apart in some cases. For example:
Distance (traveled) = Time in transit × Speed, or D = T × S,
where S is the speed of movement. Is that an empirical statement or a definition?
Either way, one can manipulate the formula and derive other relationships that are “necessary,” that is, compelled by or implicit in the original definition. So,
Speed = Distance divided by Time, and
Time (traveled) = Distance divided by Speed. It is apparent that these simple manipulations of the formula generate new statements of the relationship defined that can be useful.
Again, one might pause to ask whether these equations are “factual”—do they make contingent statements about the physical world? You might immediately say “of course.” If one travels for T minutes at S speed, one will in fact travel D distance. Agreed. But, is that a factual statement? Or, is it merely a definition of D (and of T and S) and, as a definition, is true because it is defined so, not because of inherent characteristics of the physical world?
Moving onward, mathematical logic tells us that if you do the same thing to both sides of an equation, then the two sides will still be equal. So, you can add the same value to both sides or subtract the same value, or multiple or divide by the same value, and the equality will still hold. Why is this proposition “true”? Because it is inherent in the definition of “equals.” But, that fact does not make the proposition vacuous.
Indeed, surprisingly, there are many examples where this process of performing the same operation on both sides of an equation, followed by manipulations of the resulting equation according to the established rules, results in a statement (in the form of something “=” something else) that seems to provide new, unexpected information about the physical world. We can generate propositions