Honors 213
Mathematicians write proofs. In these proofs the author is trying to prove the validity of a conditional statement (cstatement) of the form IF H THEN C (abbreviated as H® C) where H and C are statements called the hypothesis and conclusion respectively. For example, IF x2-4 = 0 and x < 0 THEN x = -2, is a cstatement. Cstatements are an invention of logic and a cstatement is given the value ``true'' only when C is true every time that H is true (or if H is always false). If it is possible that H is true but C is false, then the cstatement is given a value of false. When a mathematician actually tries to prove that H® C, he is attempting to deduce that if H is assumed to be true then C is also true. The way that logic helps the mathematician prove H® C is by the logical rule that if H® A and A® C then H® C. So the mathematician is always stringing together conditional statements in order to get from hypothesis to conclusion.
However the truth of cstatements, which all proofs look to validate, has nothing to do with the truth of their component statements. For example it might be the case that IF x2-4 = 0 and x < 0 THEN x = -2 is a true cstatement, however we have no reason to believe that the component statement x2-4 = 2 and x < 0 is true. If the component statement is never true (maybe x is always 7), then the cstatement is pretty useless. Unfortunately logic will not tell us whether or not it is possible for the hypothesis to be true. Additionally, for most proofs, logic is not complete enough to create the conditional statements needed to string together H and C. Therefore mathematicians must create statements that are ordained to be true a priori. These statements are called axioms and are chosen so that they are consistent (i.e., no single derivable statement is both true and false). The axioms that are chosen define the type of cstatements that can be proven and are phrased in terms of undefined terms that are particular to the subject. For example, in geometry the axioms deal with relationships between undefined terms called ``lines'' and ``points'' whereas in algebra they deal with the existence and properties of things called ``numbers''. So the conditional statements in geometry have a lot to do with lines and points whereas an algebraic conditional statement speaks about numbers. Finally, the mathematician also creates definitions and notations which are used as a means to describe complicated situations using one word or symbol. For example a triangle might be defined as three lines that enclose an area, and might be notated by D. Additionally, definitions are only made when the item which is being defined exists due to the axioms. For example our explanation of a triangle would not be a valid definition if one of our axioms implied that only two lines exist.
With the axioms and definitions in hand, a mathematician should be able to prove many cstatements. These cstatements are then considered to be truth (with respect to the particular axiomatic system). Since the cstatements depend on logic and axioms, it is tempting to say that truth in mathematics depends on the truth of logic and axioms. However, I believe that assigning truth to a set of axioms is too subjective to be considered mathematical and that pure mathematics simply has to do with the logical consequences of any consistent set of axioms. For example many might not think that hyperbolic geometry is truthful since it contains an axiom that allows more than one line parallel to any given line. However, I believe (and most mathematicians would agree) that hyperbolic geometry is a valid mathematical structure since its cstatements follow from logic and a set of consistent axioms: it is mathematical truth.
Moving on to the idea of truth in physics, we start with the statement that physicists create theories that describe how the universe operates. These theories are always mathematical in that they use mathematical cstatements and therefore adhere to the axiomatic structure of mathematics. For example, Newton's Laws can be expressed as a set of differential equations that hypothesize the relationship between force and acceleration in bodies with mass. This theory is mathematical in that it uses the axiomatic system of calculus, but is distinct from mathematics in that it presents itself as a way to understand how real things in the universe operate. In order to further illustrate what a physicist does and draw more parallels with mathematical truth, it will be beneficial to break physics into experimental physics and theoretical physics.
Experimental physicists work directly with unknown phenomena and try to show that these phenomena are describable using certain mathematical relations. Usually an experimental physicist will work on a particular experiment for a long time and attempt to describe it mathematically. An example of experimental physics is the current work to describe the time evolution of the gravitational constant in Newton's Laws. This time evolution is something that many physicists think might happen, and the experimental physicists are spending a good deal of time setting up an experiment that will test whether the time evolution exists. If they find that there is a time evolution, it will be expressible in a mathematical way (as a function). If they do not find a time evolution, they will not have proven that it does not exist, rather they will have shown that if it does exist, its value is too small to be measured by their experiment. The main requirement to establish the validity or truth of a result in experimental physics is that the result must be reproducible. This means that if other people were to do the same experiment, they would get the same result as the original experimenter. So truth in experimental physics is the reproducible phenomena that are observed and mathematically described.
Theoretical physicists do not work directly with physical phenomena. Rather they act a lot like mathematicians in that they attempt to describe a set of experimental data by creating physical axiomatic systems. For example Non-Relativistic Quantum Mechanics, a physical theory that describes the behavior of relatively slow small particles (atoms, molecules, electrons, etc.), relies on three physical axioms: conservation of energy, the form of the energy operator, and the interpretation of the psi function (Weyl, 51-56). All other axioms in the theory are the mathematical axioms that allow use of statistics, calculus, and other mathematical structures. With these physical and mathematical axioms, many predictions on how a certain particle will act in a certain situation can be made and tested by experiment. A prediction is considered true if all aspects of the prediction agree with experiment (within the error of the experiment). However it is much more difficult to verbalize when a theory is considered true in physics.
As we have seen, a physical theory is based on both mathematical and physical axioms. To build on these axioms, the physicist uses mathematical theories that have already been built up from the mathematical axioms (i.e., geometry, calculus) in order to predict phenomena. The main difference between mathematics and theoretical physics is that the mathematical symbols are given precise meaning by the physicist whereas they are seen as arbitrary by the mathematician. For an example, we can use Einstein's famous equation E = mc2 which creates an equivalence between energy (E) and mass (m). A mathematician would look at this equation and understand that there are three variables in the equation and would know how to manipulate the equation (maybe in order to solve for m). A physicist would know how to do all of the things that the mathematician could do, but would also place a meaning on the variables. Instead of the equation being an arbitrary set of variables, the physicist recognizes them as physical quantities that can be measured. In this way the mathematics is applied to the physical world, and the way that it is applied depends on the physical axioms of the system.
As an example of applying physical axioms, we will now build non-relativistic quantum mechanics from its three physical axioms (conservation of energy, the interpretation of the psi function, and the form of the energy operator) in order to get a better idea of how the process takes place. Non-relativistic quantum mechanics is built from the physical axioms by starting with the conservation of energy equation
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which states that the energy E can be written as the sum of the kinetic and potential energy. The psi function (Y) which is interpreted in the axioms as a probability density, is multiplied on each side. Then the final axiom states that the energy can be written as E = i(h/2p) [(¶)/(¶t)]. Using this expression for energy, we can also deduce that p = -i(h/2p) [(¶)/(¶x)] in one dimension. So when this last axiom is added, we can write
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which is the Schrodinger equation in one dimension. All predictions in non-relativistic quantum mechanics are made using this equation. Whether or not the reader followed this derivation (or even understood the symbols), it was inserted in order to show the process of how the theoretical physicist invents physical axioms in order to end up with an equation that relates helpful variables. The Schrodinger equation will allow a physicist to solve for the variable Y under certain situations. Through the interpretation of this variable (from the axioms) the physicist can then locate approximately where a particle is and find its energy. So then the main difference between theoretical physics and mathematics is the inclusion of physical axioms that give the mathematical variables meaning. Likewise, the truth that is associated with a theory in physics is based on the truthfulness of the physical axioms that go along with the theory. Since the physical axioms determine the predictions that the theory makes, the truest test of a physical theory is that what it predicts actually happens. If everything that a certain theory predicts can be verified experimentally, then a theory is true.
Of course this verbalization of what a physical theory must is still allows the physical axiomatic system with one axiom: the sun rises each day. Now this physical axiomatic system will only describe one phenomenon (and will only be experimentally verifiable in some places). So the second thing that physicists like in a theory is that it explains many phenomena. It is still the case that if everything a certain theory predicts can be experimentally verified, then it is true; however a theory that explains one thousand phenomena is thought better than a theory that explains two.
Through the work of experimental and theoretical physicists, truth is found by continuously inventing theories and testing them with experiment. If a theory predicts many true phenomena seen experimentally, then the theory is true. If an experiment is reproducible, then the phenomenon recorded is true. However the one thing that physicists can never do is prove that their theories or experiments will always be true. Just because a theory's predictions have been right in the past does not necessarily mean that they will be right in the future. Similarly, just because an experiment has been reproduced successfully twenty times does not mean it will be reproducible one hundred times. So the main obstacle that physicists face is that their results are never final. Even if a theory were devised today that explained all known experimental phenomenon, it would most likely be found in the future that the theory is false (some new experiment would be made). However, it is the case that if one theory is found to be false by a particular experiment, the theory that ends up replacing it is always quite similar to the original theory in some important way. So even though physicists can never say that their work is entirely done, they generally do not have to invent totally new theories every time new experimental data is found.
To summarize, a mathematical theory is true if its conditional statements follow from logic and a set of consistent axioms. A physical theory is true as long as all its known predictions are verified, and a physical experiment is true if it is reproducible. In terms of which truth I prefer, I must say physical truth. Although both physical theorists and mathematicians try to prove theorems, the physical theorist can always compare these theorems to experiments in order to see if they are correct. A mathematician can only rest assured in the fact that the logic inherent in the theorem is correct. Since the truth of the results of a physical theorem is directly associated with the truth of the physical axioms, every time a physical result is experimentally verified, the truth inherent in the axioms becomes more entrenched. So if it were possible to create a perfect theory of physics that was always true, its axioms would be true statements about the universe. This is the added dimension that physics has and mathematics does not. In mathematics you can make up any consistent axiomatic system, derive results from it, and the resulting mathematical system will be true. In physics you have the added condition of having to divine axioms. This adds fun to the game.
Duhem, Pierre. The Aim and Structure of Physical Theory. Princeton University Press, 1954.
Leplin, Jarrett. The Creation of Ideas in Physics. Kluwer Academic Publishers, 1995.
Weyl, Hermann. The Theory of Groups and Quantum Mechanics. Dover Publications, 1931.