In his chapter on positivism, Richard Fuerle writes:
A deductionist would verify the geometrical law that all triangles have 180° (i.e., if any angle of a triangle changes, one or both of the other angles must also change so that they sum to 180°) by placing a line parallel to one of the sides of any triangle at the apex of the other two sides.
<------ new line
<------ triangle
<------ base
Since the internal opposite angles formed when a line crosses parallel lines are equal (a’ = a and b’ = b), and a straight line is defined to be 180°, it necessarily follows that the three angles of the triangle, a, b, and c, must total 180°. The positivist, however, would attempt to verify that geometric law by measuring the angles of thousands of different triangles. He might arrive at the conclusion that the angles total 180° within a probable error of ± 0.01 percent, but he could never prove that the answer was exactly 180°, and he could never prove that the next triangle he measured must total 180° and not an entirely different number (p. 53).
I agree with Fuerle’s condemnation of positivism. Elsewhere on this website, I have written a paper, Socrates and Hume at Billiards, comparing how a positivist and a physicist would answer a question about where a billiard ball was going to go after being struck by the cue ball. I have Socrates and Hume decide in favor of the physicist, not the positivist. However, as should be clear from reading Socrates and Hume at Billiards, I do not agree with Fuerle when he writes:
I can not be certain that every physical law holds true everywhere and at all times, but, given necessary conditions, I am certain that the laws of economics, like the laws of geometry, do. It is the view that economics is an empirical science like physics that is responsible for the disarray and error that is so widespread in the discipline, and the quite justified contempt that members of the general public have for economics (p. xv).
I will not deny that disarray and error are widespread in economics, but I will deny that physics is an empirical science. Geometry, physics and economics are all axiomatic. How could such important fields of inquiry differ on so basic an issue? But this does not mean that their laws hold true everywhere and at all times. An axiom is a proposition that is assumed without proof for the sake of studying the consequences that follow from it. Whether one’s axioms are wisely chosen in the sense that they have any application to the real world is an entirely separate question from whether one’s theorems actually follow from one’s axioms.
The two axioms discussed in my paper, Socrates and Hume at Billiards, are that momentum and energy are both conserved. I think everyone will agree that these axioms were wisely chosen. Neither axiom has let us down in the hundreds of years since Newton founded modern physics. The three axioms discussed in my book, Axiomatic Theory of Economics, are that value conforms to the definitions of total ordering, marginal utility and proportionate effect. I think that these axioms are also wisely chosen, though others may disagree. People may, though nobody has yet, also find fault with the proofs of my theorems, feeling that my theorems do not actually follow from my axioms. Hopefully, it will eventually be agreed that my axioms, like Newton’s axioms, are wisely chosen and also that my theorems are soundly proven. Right now, however, the only point I wish to make is that these are two separate questions.
While Fuerle’s purpose in writing the passage quoted above is to condemn positivism, it is actually the first part of this passage, “a deductionist would verify the geometrical law that all triangles have 180° by placing a line parallel to one of the sides of any triangle at the apex of the other two sides,” that I take issue with.
Recall that the term “deduct” means to remove something. Yet the first thing Fuerle’s deductionist did was to add something. He added the new line at the top of the figure which touches the apex and is parallel to the base. How can this process be called deduction if it requires adding something new? Yet the result, that the interior angles of a triangle sum to 180°, refers only to the original figure, the triangle, and makes no mention of the new line. So, apparently, this new line was added and then deducted, though Fuerle does not explicitly say so.
In fact, this is an example of what I refer to as synthetic a priori knowledge. In my book, Axiomatic Theory of Economics, I write:
“While there are a finite number of theories implied by a given one, there is an infinite number of theories that could have implied a given theory. These theories are found by adding definitions (which are always available) to an alternative of the given theory. If, out of this infinity of theories, the analysis of one of them yields an alternative that contains only the characteristics of the given theory (which is implied by the one under analysis) and yet imply a relation that is not contained in that given theory, this relation is synthetic a priori knowledge...
“The use of additional definitions which are then deducted after a solution has been found is often forgotten, leading people to believe that synthetic a priori knowledge is impossible and that all understanding is analytic. That synthesis is a passing event which leaves no mark on its creation and that all declarative sentences are analyzable from discursive postulates has led many linguists to take this stand...
“An illustration of synthetic a priori knowledge will now be given:
“If definition p is of a given equilateral triangle XYZ and definition q is of a square, what additional definition(s) r must be added to pq so that the analysis of pqr leaves only the characteristics of p and q (there are no superfluous lines) but with the square given definite size, ABCD, so that it fits inside triangle XYZ, as shown in this figure?
“Neither the definition of p (three equal sides of definite length) nor the definition of q (four equal sides of indeterminate length) contains any information about the position of B or C. Additional characteristics are needed (lines have to be drawn) to find these lengths. But these additional characteristics have to be deducted again for the new theory to keep the same extension...
“This Figure illustrates the solution. This figure, pqr, is an anti-implication of pq, as it can imply pq by deducting the additional lines just added. They are not needed for phenomena to conform to the square inside the triangle. After their deduction, however, the relation of B and C to lines XY and YZ, which was not known before, is still there. This relation is synthetic a priori knowledge.
“A more algebraic example is the integration of . The first three steps establish the needed anti-implication.
“Mathematicians will confirm that much of what they do involves finding a ‘trick’, usually an identity, which transforms a given equation into one which superficially seems more complicated but which in fact easily implies what they are trying to prove. Almost invariably, once this trick is discovered, the rest of a proof is just a matter of algebraic manipulation (pp. 43-48).”
Fuerle does not acknowledge the possibility of synthetic a priori knowledge in his book, The Pure Logic of Choice. Of course, neither do any other Kantian philosophers, so it would be unfair to single Fuerle out for special condemnation. Nevertheless, this is the principle weakness of his epistemology.
I would, however, like to commend Fuerle for making an effort to establish a foundation for his science before plunging directly into applications, which is the more common approach. He writes (p. xvi), “When Alice asked the Queen of Hearts where to begin, she was told to begin at the beginning. The beginning is epistemology – the study of how to validate and verify knowledge – and ontology – the study of the nature of those things about which we wish to acquire knowledge. Most [economists] have not begun at the beginning.” Fuerle did begin at the beginning and kept at it for a good 80 pages, which is commendable, if a bit off-putting.
Furthermore, Fuerle writes (p. xvii), “While von Mises must be given credit for generalizing the subjective, deductive approach to economics into an overall theory of human action, von Mises rejected the idea of presenting his theory in the form of universal axioms, preferring instead to begin with the common experience of all mankind.” I agree. Mises spent a lot of time talking about deduction, but very little time actually doing deduction. His “action axiom” is really just a platitude. As I ask in my Critique of Austrian Economics, “Whoever heard of an axiomatic system with only one axiom? There are only postulate sets (e.g. Euclid has five, Kolmogorov has five and this author has three) (p. 34).”
Unfortunately, Fuerle confuses his conception of epistemology – the study of how to validate and verify knowledge – with the invention of new theories. He takes what I refer to in my book as the linguist’s approach:
As linguists deal with theories whose creation has been forgotten and which have turned into statements that could have as easily been handed down from a mountain as synthesized, it is not surprising that they should regard these as cases of analytic knowledge. They need only ask "What do the words mean in this configuration?" and they know the meaning of the theory. They forget that at one time the theory was unknown but a simpler one was known without certain relations. Then people noticed that, whenever they used the theory, the phenomena that conformed to it had those relations, but they were hesitant to risk anything on the assumption that future phenomena would have those relations also, for they could not be sure that it was not a coincidence. Then someone found an anti-implication of the theory which, when analyzed, yielded those relations as synthetic a priori knowledge, so people no longer had to wonder if the next phenomenon that conformed to their theory would have those relations but could relax and say "Whatever phenomena conform to these characteristics has these relations." Or it might have happened another way and it was some of the characteristics which people were unsure of and someone found an anti-implication of the known relations that, when analyzed, yielded those characteristics as synthetic a priori knowledge. But that has all been forgotten and now linguists only see a theory with certain characteristics and certain relations, so they analyze it and then proclaim that synthetic a priori knowledge is impossible (pp. 44-45).
The difference between the linguist and the positivist is in their method, not their approach. Both the linguists and the positivists are dealing “with theories whose creation has been forgotten and which have turned into statements that could have as easily been handed down from a mountain as synthesized” and are interested in validating and verifying them. The difference is in how they go about verification of the theories they are given.
The linguist analyzes the meaning of the words and symbols in the result. For instance, if given the statement
he would inquire about the meaning of the symbols ln (natural logarithm), (infinity), ! (factorial) and Σ (summation) in order to know what the formula means. The positivist would insist on evaluating the integral to determine the area from, say, one to ten, and then comparing the result to the output of a numerical integration program which divides the area under the graph into a number of little rectangles and then adds them up.
Clearly, the positivist’s approach has weaknesses. He can divide the area under the graph into a couple dozen or even a hundred little rectangles, but he cannot divide it into an infinity of rectangles. Neither can he evaluate more than the first dozen or so terms of the infinite summation. So there will always be some uncertainty in his conclusion. Not only that, but how can he be sure that the accuracy he measures when integrating from one to ten is comparable to the accuracy that will be realized by someone who needs to know the area under the graph from, say, five to twenty?
The linguist’s approach is relevant because, if one does not know what the symbols represent, one has no way of evaluating the formula. But neither the linguist nor the positivist have answered the most important question of all: If one is presented withon a calculus test, how is one supposed to respond?
The professor is not going to accept a computer printout of a numerical integration program and telling him what natural logarithms are is not going to win many points. The bottom line is, how is one supposed to figure out how to integrate ?
It is this and similar questions that I hope to answer, at least in a general way, with my discussion of how synthetic a priori knowledge is possible.
Click here for a side-by-side comparison of the Austrian and Axiomatic theories' foundations.
. The first three steps establish the needed anti-implication.
(infinity), ! (factorial) and 
on a calculus test, how is one supposed to respond?