





































Advances in Politics and Economic 

ISSN 2576-1382 (Print) ISSN 2576-1390 (Online) 

Vol. 2, No. 3, 2019 

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229 
 

Original Papers 

The Role of Ignorance about Keynes’s Inexact, Approximation 

Approach to Measurement in the A Treatise on Probability in the 

Keynes-Tinbergen Exchanges of 1938-1940 in the Economic 

Journal 

Michael E Brady1*∙ 

1 Department of Operations Management, College of Business Administration and Public Policy, 

California State University, Dominguez Hills, 1000 Victoria St, Carson, California, 90747, USA 

* Michael E Brady, Department of Operations Management, College of Business Administration and 

Public Policy, California State University, Dominguez Hills, 1000 Victoria St, Carson, California, 

90747, USA 

 

Received: July 28, 2019        Accepted: August 6, 2019      Online Published: August 19, 2019 

doi:10.22158/ape.v2n3p224           URL: http://dx.doi.org/10.22158/ape.v2n3p229 

 

Abstract 

J.Tinbergen and J. M. Keynes held diametrically opposed positions on measurement. Tinbergen’s 

physics background led him to deploy an exact approach to measurement based on the specification of 

probability distributions, like the normal and log normal, with exact and precisely probabilities that 

were linear, additive and definite. All probabilities for Tinbergen were assumed to be well defined, 

precise, exact, determinate, definite, additive, linear, independent single number answers, whether the 

field was physics or economics. Keynes’s approach was an inexact one. Probabilities for Keynes were, 

with a few exceptions, partially defined, imprecise, inexact, indefinite, indeterminate ,non additive, non 

linear, and dependent. Probability estimates for Keynes required two numbers to specify the probability 

within a lower and upper bound (limit), and not one single number like Tinbergen’s approach. Keynes 

called this approach Approximation. Keynesian probabilities are interval valued. 

The problem, from Keynes’s perspective, was that Tinbergen was trying to apply to economic data 

techniques which were only sound when applied in physics ,where laboratory controlled environments 

with detailed experimental design could generate data and replicate/duplicate the experiments. Keynes 

had always argued that economics was not a physical or life science like physics, engineering, biology 

or chemistry and that it could never be like physics. 



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Keywords 

exact measurement, inexact measurement, precise probability, imprecise probability, approximation, 

interval valued probability 

 

1. Introduction 

The paper will be organized in the following fashion. Section Two will present an overview of 

Keynes’s inexact approximation approach to measurement as presented by him in 1921 in the A 

Treatise on Probability (TP, 1921). All five parts of the TP will be examined in order to show that 

Keynes’s concern with the inexact measurement of probability in statistics was one of the primary 

goals Keynes had in writing the book. Section Three will examine the mistaken assessments made by 

Tinbergen and other economists about Keynes’s technical understandings of probability and statistics. 

It will be demonstrated that Tinbergen and other economists were advocating techniques for application 

in economics that worked only in physics. Section Four concludes the paper. 

Keynes’s unchanging lifetime views on inexact measurement versus exact measurement and imprecise 

probability versus precise probability explain the Keynes-Tinbergen debate of 1939-1940. Tinbergen’s 

background in Physics meant that he was an advocate of the Limiting Frequency interpretation of 

probability. Tinbergen was thus an advocate of precise and definite probability and exact measurement. 

Tinbergen brought this view with him when he started working in economics. Keynes was the exact 

opposite. 

Tinbergen was used to analyzing inanimate phenomenon, like atoms, protons, electrons, particles, cells, 

molecules, genes, chromosomes, fair decks of cards, dice, coins, etc. Keynes was used to analyzing 

animate human behavior. Humans, unlike inanimate phenomenon, like atoms, protons, electrons, 

particles, cells, molecules, genes, chromosomes, fair decks of cards, dice, coins, etc., can think and 

reason. Humans have brains, memories, emotions, minds, can reason and change their opinions. While 

exact calculation and precision can be applied in many areas of the physical and life science, it is 

doubtful in most social sciences, liberal arts, behavioral sciences and especially in most areas of 

economics, finance and business. 

 

2. Method 

It is impossible for technically trained reader of the TP not to recognize that Keynes’s method is one of 

inexact measurement based on approximation unless the reader has not read Parts II and V. 

Mathematically confused readers of the TP would not be able to understand Keynes’s use of Boole’s 

concept of lower and upper ,interval valued, probability that was used by Keynes in parts I, III and IV, 

and heavily by Keynes in Parts II and V of the TP. Mathematical confusion may be the best explanation 

available that explains the incoherent, incomprehensible, contradictory Keynesian Fundamentalist 

claim that Keynes’s method was one of ordinal measurement some of the time. Ordinal measurement 

appears nowhere in the TP. This section of the paper will be divided into seven subsections 2.1 to 2.7.  



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2.1 Keynes’s Inexact Measurement and Approximation Method in Part I of the TP 

Keynes refers to inexact measurement (intervals) a number of times in chapter III of the TP. His main 

target is the claim that all probabilities are numerical. In Part II, he shows that this would be the 

assertion that all probabilities are linear and additive. He warns the reader explicitly on page 37 that his 

discussions in chapter III are preliminary only and that it is in Part II that the issue of numerical 

measurement will be discussed in detail: “It will not be possible to explain in detail how and in what 

sense a meaning can sometimes be given to the numerical measurement of probabilities until Part II. is 

reached. But this chapter will be more complete if I indicate briefly the conclusions at which we shall 

arrive later…” (Keynes, 1921, p. 37; boldface added). Thus, Keynes’s view of (pp. 38-40) is that they 

provide a “brief”, but not “detailed”, set of conclusions ahead of the main conclusions in Part II .It is in 

Part II ,and not in chapter III of Part I, where Keynes’s provided the details concerning the question of 

measurement. Keynes’s answer is that one needs to use approximation, using inexact measurement, 

involving Boolean interval valued probability. All Keynesian fund amentalists (Skidelsky, Carabelli, 

O’Donnell, Fitzgibbons, Runde, Meeks, etc.,), just Borel (1923), Edgeworth (1922), Ronald Fisher 

(1923), E. Wilson (1923; Pearl, 1922), and Ramsey skipped Part II of the TP. They thus arrived at a 

very incomplete answer as regards Keynes’s theory of measurement. Their incomplete answer resulted 

in their reading into pp. 38-40 of the TP an ordinal theory of probability that does not exist in the TP. 

Nowhere in the TP does Keynes put forth an ordinal (ordinary language) theory of probability or even 

list the words “ordinal”, “comparative”, or “rank-order” in his index. The diagram that Keynes presents 

the reader on p. 39 (p. 42 of Volume 8 of the CWJMK) involving pages 38-40 is a brief introduction to 

his theory of Measurement as Keynes stated on pp. 37-38 of the TP. Chapter III is the alpha of Keynes’s 

analysis of measurement. Chapters 15-17 are the omega of Keynes’s analysis. The Keynesian 

“fundamentalists” (G. Meeks, R. Skidelsky, R. O’Donnell, A. Carabelli, A. Fitzgi-bbons, J. Davis, J. 

Runde, for example) chose to ignore Part II of the TP and base all of their analysis on the diagram on 

page 39 on chapter III of Part I of the TP alone. As Keynes stated on pp. 37-38, it will not be possible to 

understand his position on the measurement of probabilities without the important ,detailed analysis of 

Part II of the TP. Keynes is very clear that probabilities can be estimated using lower and upper bounds: 

“A relation of probability does not yield us, as a rule, information of much value, unless it invests the 

conclusion with a probability which lies between narrow numerical limits. In ordinary practice, 

therefore, we do not always regard ourselves as knowing the probability of a conclusion, unless we can 

estimate it numerically. We are apt, that is to say, to restrict the use of the expression probable to these 

numerical examples, and to allege in other cases that the probability is unknown. We might say, for 

example, that we do not know, when we go on a railway journey, the probability of death in a railway 

accident, unless we are told the statistics of accidents in former years; or that we do not know our 

chances in a lottery, unless we are told the number of the tickets. But it must be clear upon reflection 

that if we use the term in this sense,—which is no doubt a perfectly legitimate sense,—we ought to say 

that in the case of some arguments a relation of probability does not exist, and not that it is unknown. 



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For it is not this probability that we have discovered, when the accession of new evidence makes it 

possible to frame a numerical estimate. Possibly this theory of unknown probabilities may also gain 

strength from our practice of estimating arguments, which, as I maintain, have no numerical value, by 

reference to those that have. We frame two ideal arguments, that is to say, in which the general 

character of the evidence largely resembles what is actually within our knowledge, but which is so 

constituted as to yield a numerical value, and we judge that the probability of the actual argument lies 

between these two. Since our standards, therefore, are referred to numerical measures in many (author- 

note Keynes’s emphasis in italics) two numerical measures, we come to believe that it must also, if 

only we knew it, possess such a measure itself”. (Keynes, 1921, pp. 31-32; bold face added). This 

quotation is the foundation for F. Y. Edgeworth’s conclusion in his 1922 review that Keynesian 

probabilities are intervals. Keynes also explicitly discusses intervals on pp. 22, 23, 24, 29, and 35 of 

chapter III. Keynes’s diagram on page 39 (p. 42 of CWJMK version of the TP) presents no theory of 

ordinal probability just as chapter 3 of the GT contains no theory of expected aggregate demand and 

expected aggregate supply.  

2.2 Keynes’s Method of Inexact Measurement and Approximation in Part II of the TP  

Keynes’s characterization of his approach again emphasizes the importance of probabilities being 

“between” an upper bound and a lower bound: “It is evident that the cases in which exact numerical 

measurement is possible are a very limited class, generally dependent on evidence which warrants a 

judgment of equi-probability by an application of the Principle of Indifference. The fuller the evidence 

upon which we rely, the less likely is it to be perfectly symmetrical in its bearing on the various 

alternatives, and the more likely is it to contain some piece of relevant information favouring one of 

them. In actual reasoning, therefore, perfectly equal probabilities, and hence exact numerical measures, 

will occur comparatively seldom. The sphere of inexact numerical comparison is not, however, quite so 

limited. Many probabilities, which are incapable of numerical measurement, can be placed nevertheless 

between (Author-note Keynes’s emphasis) numerical limits. And by taking particular non-numerical 

probabilities as standards a great number of comparisons or approximate measurements become 

possible. If we can place a probability in an order of magnitude with some standard probability, we can 

obtain its approximate measure by comparison. This method is frequently adopted in common 

discourse” (Keynes, 1921, p. 160).  

Pp. 161-163 are of great importance as Keynes demonstrates his method of analysis using 

approximation, as he does also in chapter 17 on pp. 186-194, pp. 235-238, and pp. 253-255. Keynes’s 

presentation of his method of inexact measurement and approximation in Part II can only lead to the 

complete and total rejection of the fundamentalist claim that purports to have discovered Keynes’s 

“theory” on pages 38-40 of the TP in an analysis that Keynes calls “brief”.  

2.3 Keynes’s Method of Inexact Measurement and Approximation in Part III of the TP 

On page 192 of the TP, when applying his technique for solving Boolean interval probabilities Keynes 

used this technique in part III in order to set up an application of his concept of finite probability: 



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“There is one class of probabilities, however, which I called the numerical class, the ratio of each of 

whose members to certainty can be expressed by some number less than unity; and we can sometimes 

compare a non-numerical probability in respect of more and less with one of these numerical 

probabilities. This enables us to give a definition of “finite probability” which is capable of application 

to non-numerical as well as to numerical probabilities. I define a “finite probability” as one which 

exceeds some numerical probability, the ratio of which to certainty can be expressed by a finite number. 

The principal method, in which a probability can be proved finite by a process of argument, arises 

either when its conclusion can be shown to be one of a finite number of alternatives, which are between 

them exhaustive or, at any rate, have a finite probability, and to which the Principle of Indifference is 

applicable; or (more usually), when its conclusion is more probable than some hypothesis which 

satisfies this first condition” (Keynes, 1921, p. 237; bold face added).  

Keynes repeats himself in chapter 22:  “There is a vagueness, it may be noticed, in the number of 

instances, which would be required on the above assumptions to establish a given numerical degree of 

probability, which corresponds to the vagueness in the degree of probability which we do actually 

attach to inductive conclusions. We assume that the necessary number of instances is finite, but we do 

not know what the number is. We know that the probability of a well-established induction is great, but, 

when we are asked to name its degree, we cannot. Common sense tells us that some inductive 

arguments are stronger than others, and that some are very strong. But how much stronger or how 

strong we cannot express. The probability of an induction is only numerically definite when we are able 

to make definite assumptions about the number of independent equi-probable influences at work. 

Otherwise, it is non-numerical, though bearing relations of greater and less to numerical probabilities 

according to the approximate limits within which our assumption as to the possible number of these 

causes lies” (Keynes, 1921, p. 259). 

Of course, a reading of chapter 15 of the TP is absolutely required if a reader of Part III of the TP is to 

understand that non numerical probabilities are interval valued probabilities. There is no explicit 

evidence that the Keynesian fundamentalists understand this, given their over 40 years of claims that 

Keynes’s non numerical probabilities were ordinals. Keynes’s definition of finite probability, which is 

applicable to both numerical and non numerical, interval valued probabilities, makes no sense if 

Keynes’s probabilities are ordinal. In fact, Part III of the TP is incomprehensible, inconsistent, and 

contradictory if based on a reading that argues that Keynes’s theory of probability was an ordinal one. 

This would directly conflict with the Boolean foundations provided to Part III by Boole’s Problem X on 

page 192 in chapter 17, which is interval. There are no ordinal probability problems in Boole’s 1854 

The Laws of Thought. Any interpretation of Keynesian probabilities as being ordinal probabilities 

renders any understanding of the Boolean foundation of Part III impossible to understand.  

2.4 Keynes’s Method of Inexact Measurement and Approximation in Part IV of the TP 

Consider the necessity in Part IV of understanding Keynes’s finite probabilities from Part III: 

“For this purpose it is necessary to recur, briefly, to the analysis of Part III. It was argued there that the 



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methods of empirical proof, by which we strengthen the probability of our conclusions, are not at all 

dissimilar, when we apply them to the discovery of formal truth, and when we apply them to the 

discovery of the laws which relate material objects, and that they may possibly prove useful even in the 

case of metaphysics; but that the initial probability which we strengthen by these means is differently 

obtained in each class of problem. In logic it arises out of the postulate that apparent self-evidence 

invests what seems self-evident with some degree of probability; and in physical science, out of the 

postulate that there is a limitation to the amount of independent variety amongst the qualities of 

material objects. But both in logic and in physical science we may wish to consider hypotheses which it 

is not possible to invest with any à priori probability and which we entertain solely on account of the 

known truth of many of their consequences. An axiom which has no self-evidence, but which it seems 

necessary to combine with other axioms which are self-evident in order to deduce the generally 

accepted body of formal truth, stands in this category. A scientific entity, such as the ether or the 

electron, whose qualities have never been observed but whose existence we postulate for purposes of 

explanation, stands in it also. If the analysis of Part III. is correct, we can never attribute a finite 

probability to the truth of such axioms or to the existence of such scientific entities, however many of 

their consequences we find to be true. They may be convenient hypotheses, because, if we confine 

ourselves to certain classes of their consequences, we are not likely to be led into error; but they stand, 

nevertheless, in a position altogether different from that of such generalizations as we have reason to 

invest with an initial probability (Keynes,1921, pp. 299-300; bold face added). 

Keynes’s footnote one adds the following point that ordinal probability can’t be applied:  

“I am assuming that there is no argument, arising either from self-evidence or analogy, in addition to 

the argument arising from the truth of their consequences, in favour of the truth of such axioms or the 

existence of such objects; but I daresay that this may not certainly be the case. The reader may be 

reminded also that, when I deny a finite probability this is not the same thing as to affirm that 16 the 

probability is infinitely small. I mean simply that it is not greater than some numerically measurable 

probability” (Keynes, 1921, pp. 299-300; bold face added).  

The reader should note again that this passage makes absolutely no sense if Keynes’s probabilities are 

ordinal, since finite probabilities must apply to both numerical and non numerical interval valued 

probabilities. Consider chapter 26, which was viewed as being very important by both Edgeworth and 

Russell, since the discussions in chapter 26 required a theory of inexact measurement to have been 

already presented in an earlier part of the book (Part II), according to Russell (1922):  

“In Chapter III. of Part I. I have argued that only in a strictly limited class of cases are degrees of 

probability numerically measurable. It follows from this that the ‘mathematical expectations of goods 

or advantages are not always numerically measurable; and hence, that even if a meaning can be given 

to the sum of a series of non-numerical ‘mathematical expectations,’ not every pair of such sums are 

numerically comparable in respect of more and less. Thus even if we know the degree of advantage 

which might be obtained from each of a series of alternative courses of actions and know also the 



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probability in each case of obtaining the advantage in question, it is not always possible by a mere 

process of arithmetic to determine which of the alternatives ought to be chosen. If, therefore, the 

question of right action is under all circumstances a determinate problem, it must be in virtue of an 

intuitive judgment directed to the situation as a whole, and not in virtue of an arithmetical deduction 

derived from a series of separate judgments directed to the individual alternatives each treated in 

isolation. We must accept the conclusion that, if one good is greater than another, but the probability of 

attaining the first less than that of attaining the second, the question of which it is our duty to pursue 

may be indeterminate, unless we suppose it to be within our power to make direct quantitative 

judgments of probability and goodness jointly. It may be remarked, further, that the difficulty exists, 

whether the numerical indeterminateness of the probability is in trinsicor whether its numerical value is, 

as it is according to the Frequency Theory and most other theories, simply unknown” (Keynes, 1921, p. 

312). 

Keynes has just ruled out ordinal probability due to intervals overlapping with one another meaning 

that “…not every pair of such sums are numerically comparable in respect of more and less.” It has 

been known for centuries that it is mathematically impossible to add (sum), subtract, divide, or multiple 

ordinal probabilities. Keynes’s comment can only be applied to overlapping interval valued 

probabilities. It is not possible to understand Part IV of the TP if Keynes’s theory of probability is an 

ordinal one, since finite probabilities can’t possibly be defined. Indeterminate probability means for 

Keynes, like it meant for Boole, that the addition of more relevant evidence is not going to result in a 

narrowing of the wide gap that exists between the lower and upper probabilities.  

2.5 Keynes’s Method of Inexact Measurement and Approximation in Part V of the TP 

Keynes repeatedly refers to indirect and inexact measurement using intervals all through Part V of the 

TP: The “Stability of Statistical Frequencies” would be a much better name for it. The former suggests, 

as perhaps Poisson intended to suggest, but what is certainly false, that every class of event shows 

statistical regularity of occurrence if only one takes a sufficient number of instances of it. It also 

encourages the method of procedure, by which it is thought legitimate to take any observed degree of 

frequency or association, which is shown in a fairly numerous set of statistics, and to assume with 

insufficient investigation that, because the statistics are numerous, the observed degree of frequency is 

therefore stable. Observation shows that some statistical frequencies are, within narrower or wider 

limits, stable But stable frequencies are not very common, and cannot be assumed lightly. 19 The 

gradual discovery, that there are certain classes of phenomena, in which, though it is impossible to 

predict what will happen in each individual case, there is nevertheless a regularity of occurrence if the 

phenomena be considered together in successive sets” (Keynes, 1921, p. 336). I will simply list the 

great number of statements by Keynes where frequency probabilities are reinterpreted as interval 

probabilities that lie between upper and lower limits, so that approximation applies. One of the values 

of reading Part V of the TP is that it shows the reader what the real ,underlying conflict was that existed 

between Keynes and Tinbergen over measurement versus Tinbergen. It can be seen to revolve around 



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the fact that Tinbergen was committed to Exact measurement using precise probabilities while Keynes 

was committed to Inexact measurement using imprecise probability based on his approximation 

technique from chapters 15-17 of the TP or methods like those of the statistician Yule:  

• Beginning with Bernoulli’s Theorem, we will consider the various solutions of this problem which 

have been propounded and endeavour to determine the proper limits within which each method has 

validity (Keynes, 1921, pp. 337-338). 

• …the value of this probability being calculable by a process of approximation… (Keynes, 1921, p. 

338).  

• For the second part of the theorem some method of approximation is required… (Keynes, 1921, p. 

338). 

• …by means of Stirling’s Theorem, and obtain as its approximate value…(Keynes, 1921, p. 338).  

• It is possible, of course, by more complicated formulae to obtain closer approximations than this.* 

But there is an objection, which can be raised to this approximation, quite distinct from the fact that it 

does not furnish a result correct to as many places of decimals as bit might. This is, that the 

approximation is independent of the sign of h, whereas the original expression is not thus independent. 

That is to say, the approximation implies a symmetrical distribution for different values of h about the 

value for h = 0; while the expression under approximation is unsymmetrical. It is easily seen that this 

want of symmetry is 20 appreciable unless mpq is large. We ought, therefore, to have laid it down as a 

condition of our approximation, not only that m must be large, but also that mpq must be large. Unlike 

most of my criticisms, this is a mathematical, rather than a logical, point (Keynes, 1921, pp. 338-339).  

• This “fiction” will do no harm so long as it is remembered that we are now dealing with a particular 

kind of approximation…the probability* that the number of occurrences will lie between… (Keynes, 

1921, p. 339). 

• This same expression measures the probability that the proportion of occurrences will lie between… 

(Keynes, 1921, pp. 339-340).  

• The probability that the proportion of occurrences will lie between given limits varies with the 

magnitude of the square root of 2pq/m, and this expression is sometimes used, therefore, to measure the 

‘precision’ of the series. Given the à priori probabilities, the precision varies inversely with the square 

root of the number of instances.(Keynes, 1921, p. 340).  

• Such a condition is very seldom fulfilled. If our initial probability is partly founded upon experience, 

it is clear that it is liable to modification in the light of further experience. It is, in fact, difficult to give 

a concrete instance of a case in which the conditions for the application of Bernoulli’s Theorem are 

completely fulfilled. At the best we are dealing in practice with a good approximation, and can assert 

that no realised series of moderate length can much affect our initial probability… For this is an 

approximate formula which requires for its validity that the series should be long; whilst it is precisely 

in this event, as we have seen above, that the use of Bernoulli’s Theorem is more than usually likely to 

be illegitimate (Keynes, 1921, p. 342).  



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• …the probability that the number of occurrences m of the event in the s trials will lie between the 

limits sp ± l is given by…The probability that the number of occurrences of the event will lie between  

sp ± γk (square root of s) is given by …(Keynes, 1921, p. 345).  

• It seems in plain opposition to good sense that on such evidence we should be able with practical 

certainty…. to estimate the number of female births within such narrow limits. And we see that the 

conditions laid down in § 11 have been flagrantly neglected…” (Keynes, 1921, p. 352). 

• Leibniz’s reply goes to the root of the difficulty. The calculation of probabilities is of the utmost value, 

he says, but in statistical inquiries there is need not so much of mathematical subtlety as of a precise 

statement of all the circumstances. The possible contingencies are too numerous to be covered by a 

finite number of experiments, and exact calculation is, therefore, out of the question. Although 

naturehas her habits, due to the recurrence of causes, they are general, not invariable. Yet empirical 

calculation, although it is inexact, may be adequate in affairs of practice” (Keynes, 1921, p. 368). 

• In dealing with the correspondence of Leibniz and Bernoulli, I have not been mainly influenced by 

the historical interest of it. The view of Leibniz, dwelling mainly on considerations of analogy, and 

demanding “not so much mathematical subtlety as a precise statement of all the circumstances”, is, 

substantially, the view which will be supported in the following chapters. The desire of Bernoulli for an 

exact formula, which would derive from the numerical frequency of the experimental results a 

numerical measure of their probability, preludes the exact formulas of later and less cautious 

mathematicians, which will be examined immediately” (Keynes, 1921, p. 369).  

• They showed, that is to say, that certain observed series of events would have been very improbable, 

if we had supposed independence between 22 some two factors or if some occurrence had been 

assumed to be as likely as not, and they inferred from this that there was in fact a measure of 

dependence or that the occurrence had probability in its favour. But they did not endeavour to pass 

from the observed frequency of occurrence to an exact measure of the probability. With the advent of 

Laplace more ambitious methods took the field” (Keynes, 1921, pp. 369-370).  

• Thus, given the frequency of occurrence in μ trials, these writers infer the probability of occurrence at 

subsequent trials within certain limits, just as, given the à priori probability, Bernoulli’s Theorem would 

enable them to predict the frequency of occurrence in μ trials within corresponding limits. If the 

number of trials is at all numerous, these limits are narrow and the purport of the inversion of 

Bernoulli’s Theorem may therefore be put briefly as follows. By the direct theorem, if p measures the 

probability, p also measures the most probable value of the frequency; by the inversion of the theorem 

also measures the most probable value of the probability. The simplicity of the process has 

recommended it, since the time of Laplace, to a great number of writers. Czuber’s argument, criticized 

on p. 399, with reference to the proportions of male and female births in Austria, is based upon an 

unqualified use of it. But examples abound throughout the literature of the subject, in which the theory 

is employed in circumstances of greater or less validity” (Keynes, 1921, pp. 370-371).  

• What, in the first place, does Laplace mean by an unknown probability? He does not mean a 



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probability, whose value is in fact unknown to us, because we are unable to draw conclusions which 

could be drawn from the data; and he seems to apply the term to any probability whose value, 

according to the argument of Chapter III., is numerically indeterminate. Thus he assumes that every 

probability has a numerical value and that, in those cases where there seems to be no numerical value, 

this value is not non-existent but unknown; and he proceeds to argue that where the numerical value is 

unknown, or as I should say where there is no such value, every value between 0 and 1 is equally 

probable. With the possible interpretations of\the term “unknown probability”, and with the theory that 

every probability can be measured by one of the real numbers between 0 and 1, I have dealt, as 

carefully as I can, in Chapter III. If the view taken there is correct, Laplace’s theory breaks down 

immediately. But even if we were to answer these questions, not as they have been answered in Chapter 

III., but in a manner favourable to Laplace’s theory, it remains doubtful whether we could legitimately 

attribute a value to the probability of an unknown probability’s having such and such a value. If a 

probability is unknown, surely the probability, relative to the same evidence, that this probability has a 

given value, is also unknown; and we are involved in an infinite regress” (Keynes, 1921, p. 373). 

Of course, Keynes’s approximation approach uses two of the real numbers between 0 and 1. Chapter III 

is Keynes’s introduction to his critique of exact and precise probability measurement with two 

exceptions (the POI is applicable; statistical data satisfies the Lexis—Q test for stability of frequency 

data). Chapters 10-17 of Part II contain the many details of his critique.  

• Nobody supposes that we can measure exactly the probability of an induction. Yet many persons 

seem to believe that in the weaker and much more difficult type of argument, where the association 

under examination has been in our experience, not invariable, but merely in a certain proportion, we 

can attribute a definite measure to our future expectations and can claim practical certainty for the 

results of predictions which lie within relatively narrow limits. Coolly considered, this is a preposterous 

claim, which would have been universally rejected long ago, if those who made it had not so 

successfully concealed themselves from the eyes of common sense in amaze of mathematics” (Keynes, 

1921, pp. 388-389).  

It can be concluded from the extensive material covered in this section that Keynes’s view of 

measurement was that ,in general, exact and precise probability measurement and assessments are not 

possible, in general, in macroeconomics. However, inexact and imprecise probability measurement and 

assessments are, in general, possible.  

2.6 Keynes’s Method of Inexact Measurement and Approximation in Chapters 3 and 4 of the GT 

Keynes’s views on measurement in the GT are identical to those expressed in the TP. Consider the 

following statements made by Keynes: “An entrepreneur, who has to reach a practical decision as to his 

scale of production, does not, of course, entertain a single undoubting expectation of what the 

sale-proceeds of a given output will be, but several hypothetical expectations held with varying degrees 

of probability and definiteness. By his expectation of proceeds I mean, therefore, that expectation of 

proceeds which, if it were held with certainty, would lead to the same behaviour as does the bundle of 



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vague and more various possibilities which actually makes up his state of expectation when he reaches 

his decision” (Keynes, 1936, ft.3, p. 24; boldface added).  

“On this basis an attempt is made to erect a quantitative science. But it is a grave objection to this 

definition for such a purpose that the community’s output of goods and services is a non-homogeneous 

complex which cannot be measured, strictly speaking, except in certain special cases, as for example 

when all the items of one output are included in the same proportions in another output…. The fact that 

two incommensurable collections of miscellaneous objects cannot in themselves provide the material 

for a quantitative analysis need not, of course, prevent us from making approximate statistical 

comparisons, depending on some broad element of judgment rather than of strict calculation, 

which may possess significance and validity within certain limits. But the proper place for such things 

as net real output and the general level of prices lies within the field of historical and statistical 

description, and their purpose should be to satisfy historical or social curiosity, a purpose for 

which perfect precision—such as our causal analysis requires, whether or not our knowledge of 

the actual values of the relevant quantities is complete or exact—is neither usual nor necessary. 

(Keynes, 1936, pp. 37-39; italics, underline and boldface added).  

Keynes states this again on p. 43: “It is my belief that much unnecessary perplexity can be avoided if 

we limit ourselves strictly to the two units, money and labour, when we are dealing with the behaviour 

of the economic system as a whole; reserving the use of units of particular outputs and equipments to 

the occasions when we are analysing the output of individual firms or industries in isolation; and the 

use of vague concepts, such as the quantity of output as a whole, the quantity of capital equipment as a 

whole and the general level of prices, to the occasions when we are attempting some historical 

comparison which is within certain (perhaps fairly wide) limits avowedly imprecise and 

approximate (Keynes, 1936, p. 43; italics, underline and boldface added).  

Keynes’s emphasis above in the GT, which is on inexact method of measurement and approximation, is 

identical to his TP approach which he applied throughout Parts II, III, IV and V of the TP. 

2.7 Conclusions 

Keynes’s unchanging lifetime views on inexact measurement versus exact measurement and imprecise 

probability versus precise probability explains the Keynes-Tinbergen debate of 1938-1940.Tinbergen’s 

background in Physics meant that he was an advocate of the Limiting Frequency interpretation of 

probability. Tinbergen was thus an advocate of precise and definite probability calculations and exact, 

precise measurement. Tinbergen brought this view with him when he started working in economics. 

Keynes held the exact opposite, diametrically opposed position. Tinbergen was used to analyzing 

inanimate phenomenon, like atoms, protons, electrons, particles, cells, molecules, genes, chromosomes, 

fair decks of cards, dice, coins, etc. Keynes was used to analyzing animate human behavior. Humans, 

unlike inanimate phenomenon, like atoms ,protons, electrons, particles, cells, molecules, genes, 

chromosomes, fair decks of cards, dice, coins, can think and reason. Humans have memories, emotions, 

minds, and brains. While exact calculation and precision can be applied in many areas of the physical 



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and life science, it is doubtful in most social sciences, liberal arts, behavioral sciences, and especially in 

most areas of economics ,finance and business. The exception would be studies of consumer 

consumption spending and business inventory demand, which are highly stable over the short run (1-5 

years). Irving Fisher’s belief that there was an analogy between a particle and an individual ((Fisher, 

1925, p. 85), so that a particle in mechanics corresponds to an individual in economics, only holds 

under the dubious claim that the sum of the parts is the whole, so that there is no interactions occurring 

among the parts as in a pizza that has been cut into 8 slices. The analogy breaks down completely once 

it is realized that a particle is an inanimate object while an individual is an animate object.  

Keynes was never an ordinalist. Nowhere in the TP or GT does Keynes develop, apply, advocate or 

deploy an ordinal theory of probability. The Keynesian fundamentalists have misinterpreted pp. 38-40 

of the TP by ignoring Keynes’s clear statement on pp. 37-38 of the TP that his detailed analysis on 

measurement would take place in Part II, while only a brief introduction to measurement would take 

place on pp. 38-40 of chapter III of the TP. Keynes would subsume Tinbergen’s approach as a special 

case of his more general case. There is no existing textual evidence that Tinbergen even understood 

what Keynes was talking about or what the original , Boolean approach to algebra-logic entailed.  

 

3. Discussion 

Consider the following exchange between Tinbergen and two interviewers in 1987: 

“From Keynes’s criticisms of your League of Nations first report, it seems fairly clear that he knew 

very little of the developments in econometrics over the 1920’s and 1930’s….(Magnus & Morgan, 

1987, p. 129). 

Tinbergen’s response to the comment by Magnus and Morgan was that  

“Indeed, I did feel that ,at least on certain points, he was badly informed. It was a bit strange to me 

because he had written the Treatise on Probability, so I thought he was somewhat familiar with 

statistics” (Magnus & Morgan, 1987, p. 129). 

The assessment made by Tinbergen ,Magnus and Morgan of Keynes’s knowledge of statistics is 

practically identical to the assessment made of Keynes’s knowledge of statistics in the TP by Ronald 

Fisher in his 1923 book review of the TP, which was that Keynes was not knowledgeable about the 

developments in statistics over the period 1880-1920 about the Law of Large numbers, Central limit 

theorem ,and their support for the use of the Gaussian (Normal) probability distribution. Fisher never 

grasped that Keynes was using inexact, numerical approximation using intervals in part V of the 

TP .The far weaker data generated by economic phenomena over time, when compared with that in 

physics, chemistry, and biology, meant that the Normal distribution would not be the appropriate 

distribution to consider. Keynes’s critical view of the assumption of Normality was examined by him in 

chapter 17 of the TP (1921) and was originally published in 1911 in the JRSS. Mandelbrot’s repeated 

50 years of demonstrations ,that exploratory data analysis in many areas of finance ,business, and 

economics does not support the use of a normal (log normal) distribution, only makes Keynes’s critique 



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appear more prescient (See Brady, 2016). 

Tinbergen appears to have made use of the Normal (lognormal) distribution throughout his career (See 

Brady’s 2019 contribution at SSRN on Keynes and Tinbergen) 

The problem that arises in the Tinbergen, Morgan, and Magnus assessments of Keynes’s knowledge is 

that they are based on a shocking and stunning ignorance of Keynes’s A Treatise on Probability. Neither 

Tinbergen, Morgan, or Magnus appear to understand Keynes’s position that Economics is not Physics 

and the calculation of invariant regression constants over decades, based on unchanging parameters, is 

simply an impossibility. Tinbergen appears to have been trying to derive constants from economic 

statistical data that are like the constants of ,for example, light and speed in physics. 

Tinbergen’s decision to use a multivariate normal distribution, or log normal, in his regression analysis 

meant that he had to treat the problem of technological innovation, advance, change ,and obsolescence, 

which is the major factor causing changes in the composition of durable, physical, capital producer, 

investment goods over the business cycle over time, as well as being the main focal point of concern of 

businessmen in forward looking, long run business expectations formation, as being part of the 

residuals or as some lagged function of past, previous views. Given the highly irregular pattern of 

actual changes in technology over time and their non normally distributed bunched nature, Tinbergen’s 

modeling strategy borders on the nonsensical. Keynes was very blunt in his assessment of Tinbergen’s 

backward looking views on modelling the expectations formation by businessmen. If Tinbergen is 

placing these important variables in the residual (error) term of the regression equation, then very large 

errors should appear. This conflicts with Tinbergen’s methodological goal of minimizing the size of the 

residuals (error terms). 

There are a few econometricians who have grasped Keynes’s point, even if they are not completely 

familiar with the inexact approach to measurement advocated by Keynes in both the TP and GT: 

“The empirical content of economic theories is therefore systematically lower than those from physical 

theory. Testing an economic theory in quantitative form requires the introduction of all sorts of ad hoc 

statistical or econometric modelling assumptions so as to arrive at a fully specified empirical model 

(Blommestein, 1985). This ad hoc nature of economic model building generates a significant degree of 

specification uncertainty. For example, as noted above, the empirical pricing models for structured 

products such as CDOs and CDSs is hampered by a considerable degree of specification uncertainty. 

Semantically insufficient theories make it therefore very hard to formulate reliable empirical models. In 

other words, the big problem with economic theories is not that they are too simplistic or that so-called 

“unrealistic‟ assumptions are being used, but it is their semantical insufficiency (low degree of 

testability). By not taking the specification uncertainty caused by semantical insufficiency seriously 

enough, testing competing theories becomes a wasteful exercise. At the minimum one needs 

statistically reliable econometric models, although technical virtuosity can never substitute for 

increasing the information content of economic theories .(Blommestein, 2009, p. 2) and “Most of the 

original Cowles work was based on the additional assumptions that error terms are IID (Independent 



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and Identically Distributed) and that functions are parameters are LCC (Linear with Constant 

Coefficients). Tellingly, DF refer to them as “articles of faith” that had (and continue to have) a 

considerable influence on the applied literature” (Blommestein, 2009, p. 3). “However, economics and 

physics are qualitatively different enterprises, and always will be. Economics or finance as a social 

science differs in at least two important respects from the physical sciences. First, physical theories 

often formulate scientific predictions in the form of individual events, while predictions in economic 

theories (and other social sciences) are usually specified as patterns of a certain kind or type (Hayek, 

1967; Blommestein, 1985, pp. 86-90). Second, theories of economic behaviour need to take into 

account reflexivity—the human capability for self-reference. Self-referential calculations have major 

(complicating) implications for economic explanations and predictions. For example, financial asset 

markets have a reflexive nature in the sense that prices are generated by the expectations of traders. But 

the latter expectations are based on the basis of the anticipation of others‟ expectations6 . This 

implication precludes the formation of expectations using deductive rules (Spear, 1989). In a 

fundamental way, economic issues (and social science topics in general) are harder and far more ” 

(Blommestein, 2009, pp. 3-4). 

And “Against the backdrop of this analysis, bringing together the insights from different parts of 

economics such as macroeconomics and financial economics is not necessarily the best response to the 

recent wave of criticism of academic finance. The end result [of this intended merger of theories] 

would still not go beyond disguised mathematics or sets of tautological structures. For example, the 

underlying behavioural paradigm of macroeconomic/macro finance models is that economic agents 

have rational expectations, meaning that the “representative” agent is at all times and everywhere 

completely informed, while fully understanding the complexities of the world (De Grauwe, 2009). This 

would imply that bankers and their clients at all times fully understand the complexities of the new 

financial landscape. It is difficult to reject the impression that these behavioural assumptions have been 

introduced to facilitate the underlying math and the formulation of quantitative models with 

closed-form solutions (Blommestein, 2009, pp. 5-6). Thus, Blommestein’s work, along with that of D. 

Freedman(2009), E Leamer (1983), and H. Keuzenkamp (2000),if it became dominant in econometric 

practice ,would go along way toward satisfying Keynes’s original objections about the use of 

econometrics in attempting to quantify business cycle research over time. 

 

4. Conclusions 

The main reasons for the completely divergent views expressed by Keynes and Tinbergen in their 

1938-1940 exchanges in the Economic Journal about measurement was due to their completely 

different technical backgrounds and different views about measurement being either exact or inexact. 

Keynes’s background was in logic, mathematics and philosophy. Tinbergen’s background was in 

physics. There simply was no common ground between the two academics. 

 



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Acknowledgement 

I thank the referee for his suggestions. 

 

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