Still More On Recurrence Of Truncation Without Reswitching

Figure 1: An Example of Structural Economic Dynamics

This post presents a diagram illustrating the effects of a particular kind of technical progress in a model with fixed capital. The example is one with two industries. Machines are produced in the machine industry and used in both the machine and corn industry. All production processes take a year to complete. Machines have a physical lifetime of two years. Managers of firms have a choice in each industry. The economic life of a machine can be one or two years in either industry.

This previous post specifies the technology at time zero. Coefficients of production for inputs and outputs are shown in Tables 1 and 2 in that post. Table 3 defines the four techniques, Alpha, Beta, Gamma, and Delta. Table 1 repeats that specification in a somewhat different format.

Table 1: Specification of Techniques

Technique	Economic Lifetime of Machines
	Machine Industry	Corn Industry
Alpha	One Year	One Year
Beta	Two Years	One Year
Gamma	One Year	Two Years
Delta	Two Years	Two Years

I consider perturbations of coefficients of production that define the amount of circulating capital needed to operate new and old machines, both in producing more new machines and in producing corn. This previous post defines the regions illustrated in the diagram at the top of this post. I repeat the definitions of the regions in Table 2 here. The cost-minimizing techniques, in each region, are listed in the order of an increasing rate of profits or decreasing wage.

Table 2: Overview of Regions

Region	Techniques	Notes
1	Alpha	No switch point.
2	Alpha, Gamma	Lower rate of profits associated with truncation in corn industry, greater output per worker.
3	Alpha, Gamma, Delta	Lower rate of profits associated with truncation, greater output per worker.
4	Alpha, Gamma, Delta, Beta	Recurrence of truncation in corn industry.
5	Alpha, Beta	Lower rate of profits associated with truncation in machine industry, greater output per worker.

I also consider the perturbations of relative markups, yielding a figure much like that at the top of this post. I conclude that both technical progress and changes in market power can have similar effects, in the large. In this example, both can bring about or eliminate the recurrence of the truncation of the economic life of the machine in one industry. This post further illustrates this claim about technical progress.

I consider further perturbations of coefficients of production in this post and this post. Although four coefficients of production decline with technical progress in Figure 1, I have not managed to specify a path through the additional regions in parameter space found in these later posts.

Sewer Socialism Worked In Milwaukee, Wisconsin, In The First Half Of The 20th Century

America's Socialist Experiment

Socialists have been elected mayor, to city councils, and to county commissions in the USA. Emil Seidel was the first socialist mayor in the USA. He was elected mayor of Milwaukee in 1910. Other socialists were elected to the city council. Another socialist, Victor Berger, was elected to the House of Representatives, to represent Milwaukee in Washington, DC.

By concentrating on Milwaukee, I am downplaying the extent to which socialists were elected mayor in cities in the USA. Wikipedia has a list, and so does the University of Washington. More than 130 mayors were socialists. Maybe I want to read David R. Berman's 2022 book, Socialist Mayors in the United States: Governing in an Era of Municipal Reform, 1900-1920.

Wisconsin is a democracy. Up until 1960, the mayor was mostly a socialist. Daniel Hoan, the second socialist mayor, was elected in 1916. Frank Zeider, the third socialist mayor, was elected in 1948.

The socialists focused on cleaning up Milwaukee, both figuratively and literally. For the former, they fought corruption and graft. They obtained their name, Sewer Socialists, from the latter. Introducing sanitation, electric, and power systems was an importance advance in their day.

The name, I guess, was coined by Morris Hillquit, a prominent member of the Socialist Party of America (1901 – 1972). He was tired of the boasting from Milwaukee socialists. But they had good cause to boast.

As I understand it, German immigrants, who fled to the US after participating in the failed 1848 revolutions, were an important current for the development of the sewer socialists. Nicholas Howland, in the link, makes an explicit link to Eduard Bernstein's reformism. Bernie Sanders, the former mayor of Burlington, Vermont, can be said to be in this tradition. Ruth Messinger, a former city council member in New York City, borough president of Manhattan, and co-chair of the Democratic Socialists of America is maybe another example.

Extensive Rent And Labor Values

1.0 Introduction

Do scarce natural resources provide additional difficultes for modern reconstructions of classical and Marxian theories of value? After all land can be sold or rented, and labor cannot produce more land. (I put aside Holland.)

This post presents an exposition of the theory of extensive rent, a start on examining possible difficulties. This type of rent provides the least dificulties, as I understand it, for such modern reconstructions. As usual, I present an example, close to the minimal complexity, needed to make my points. The model can obviously be generalized to include many more produced industrial commodities; many more types of agricultural commodities; and many more types of land, each specialized to support the production of one kind of agricultural commodity.

2.0 Technology

Table 1 specifies the technology for this example. Each column defines the coefficients of production for a process. For example, the only iron-producing process requires a_0,1 person-years of labor, a_1,1 tons of iron, and a_2,1 bushels of corn as inputs for every ton iron produced. I assume that each process requires a year to complete and exhibits constant returns to scale. The corn-producing processes each have an upper limit on how much corn they can produce.

Table 1: A Technology

	Iron Industry	Corn Industry
	Process a	Process b	Process c
Labor	a0,1	a0,2	a0,3
Land, Type 1	c1,1 = 0	c1,2 > 0	c1,3 = 0
Land, Type 2	c2,1 = 0	c2,2 = 0	c2,3 > 0
Iron	a1,1	a1,2	a1,3
Corn	a2,1	a2,2	a2,3
OUTPUTS	1 ton iron	1 bushel corn	1 bushel corn

I assume two types of land exist, distinguished by the processes that can be operated on them. A single corn-producing process can be operated on each type of land. Only a certain number of acres of each type of land exists. Each corn-producing process leaves the land unchanged at the end of operating the process. The given quantities of land limit how much corn can be produced. This model cannot accomodate a positive steady-state rate of growth without technical progress.

A full specification for this model should include requirements for use. I assume that the net output must be such that both types of land are farmed, but only one type is fully farmed. Two techniques for production exist, as shown in Table 2. All three processes are operated in each technique, but only one type of land is fully used.

Table 2: Specification of Techniques

Technique	Type 1 Land	Type 2 Land
Alpha	Partially farmed	Fully farmed
Beta	Fully farmed	Partially farmed

3.0 Parameters and Variables

I have already implicitly defined certain parameters above. Table 3 lists certain parameters I use in this model. Table 4 lists variables that I need. Some assumptions are imposed on the matrices A_α and A_β:

• All produced commodities are basic. Iron and corn enter directly or indirectly into the production of both commodities.
• The technology expressed by these matrices is productive. Each matrix satisfies the Hawkins-Simon condition.

Table 3: Selected Parameters

Symbol	Definition
a0, α	Two-element row vector consisting of first two labor coefficients.
a0, β	Two-element row vector consisting of first and third labor coefficients.
Aα	2x2 matrix, with columns consisting of iron and corn coefficients of production for first and second processes.
Aβ	2x2 matrix, with columns consisting of iron and corn coefficients of production for first and third processes.
d	Two-element column vector consisting of iron and corn quantities in the numeraire.

Table 4: Variables

Symbol	Definition
vα	2-element row vector of labor values when type 1 land is free.
vβ	2-element row vector of labor values when type 2 land is free.
p	2-element row vector of prices of unit quantities of iron and corn.
p1	The price of iron, in numeraire units per ton. The first element of p.
p2	The price of corn, in numeraire units per bushel. The second element of p.
rho1	The rent of type 1 land, in numeraire units per acre.
rho2	The rent of type 2 land, in numeraire units per acre.
w	The wage, in numeraire units per person-year.
r	The rate of profits.

4.0 Labor Values

Given the technique in use, how much additional labor would be employed throughout the economy if the net output was such that one additional unit of iron were produced? This is the labor value of iron, and it easily calculated in the theory. The answer to the same question for corn is its labor value.

Suppose type 1 land is free. Then labor values are:

v_α = a_{0, α} (I - A_α)^-1

Labor values, when type 2 land is free, are the corresponding Leontief employment multipliers for the Beta technique. Variations in net output require varying the amount of the land farmed on the type of land that is not fully farmed.

5.0 Prices of Production

With market prices, some operated processes will be obtaining a higher rate of profits than average, and some will be obtaining a lower rate. These variations in the profit rates are perhaps a signal to capitalists that they should disinvest in some industries or processes and increase investment in others. Models of cross-dual dynamics and other models explore these disequilibria.

Prices of production are such that these signals are absent. All operated processes obtain the same rate of profits. I assume profits, rents, and wages are paid out of the surplus product at the end of the year. The following three equations express the condition that all processes obtain the same rate of profits:

(p₁ a_1,1 + p₂ a_2,1)(1 + r) + w a_0,1 = p₁

(p₁ a_1,2 + p₂ a_2,2)(1 + r) + rho₁ c_1,2 + w a_0,2 = p₂

(p₁ a_1,3 + p₂ a_2,3)(1 + r) + rho₂ c_2,3 + w a_0,3 = p₂

The next equation expresses the condition that the price of the numeraire is unity:

p d = 1

Finally, one of the rents must be zero:

rho₁ rho₂ = 0

The last equation is a defining feature of the theory of extensive rent.

Suppose one of the types of land is rent-free. For deiniteness, let type 1 land be only partially farmed. Then the first four equations are in terms of five variables (p₁, p₂, rho₂, w, r). Just as in the case with only circulating capital, prices of production are specified up to one degree of freedom. In classical political economy, the wage is take as given.

6.0 Choice of Technique

Suppose the wage is non-negative and does not exceed a maximum defined by the technology. The system of equations for prices of production has two solutions. Each solution has the rent on one type of land set to zero. The cost-minimizing technique is the one in which the rent on the other land is positive. If, for a technique, the rent on a type of land is negative, that technique will not be adopted by capitalists. At a switch point, the rents on both types of land are zero.

But the analysis of the choice of technique can be expressed in terms of wage curves. Suppose rents were zero. Consider the first two equations in the system of equations for the prices of production and the equation setting the price of the numeraire to unity. These equations yield a function in which the wage decreases with an increase in the rate of profits. Similarly, the first and third equations yield another decreasing wage curve.

In the case of circulating capital alone, the cost-minimizing technique is found by the wage frontier formed out of the outer envelope of these wage curves. At a given wage, the cost-minimizing technique maximizes the wage.

In this example of extensive rent, the cost-minimizing technique is found by the wage frontier formed out of the inner envelope of the wage curves.

In either case, the appropriate wage frontier shows that a lower rate of profits is associated with a higher wage and vice versa. The maximum wage occurs when the rate of profits is zero. The maximum rate of profits arises when the wage is zero.

7.0 Special Cases

Which land is free and which land pays a rent depends on either the wage or the rate of profits, whichever is taken as exogenous in the system of prices of production. At any rate, a wage frontier exists in which the wage is higher the smaller the rate of profits. This frontier is not the outer frontier of the wage curves for the technique.

Without loss of generality, suppose the Alpha technique is cost-minimizing. Type 1 land is not fully farmed and pays no rent. Then labor values are defined, based on the iron-producing process and the process on type 1 land.

Consider the special case in which a_{0, α} is an eigenvector corresponding to the maximum eigenvector for A_α. Then relative prices of production are equal to relative labor values.

On the other hand, suppose that the numeraire is the standard commodity, as found from a_{0, α} and A_α. Suppose only the standard commodity is produced. In this case, only the process on the rent-free land would be used, in contradiction to the analysis of the choice of technique. And suppose the wage is paid out in the form of the standard commodity. Then the following hold:

• The labor value of gross output is equal to total gross output, evaluated at prices of production.
• The labor value of net output is equal to net output, evaluated at prices of production.
• The labor value of the proportion of the standard commodity paid out in wages is equal to wage goods, evaluated at prices of production.

This special case seems especially forced in the case of extensive rent. Is some reformulation available in which surplus value can be treated as the sum of profits and rent?

I do not address the use of labor values in Marx's account of exploitation, Marx-biased technical change, and so on. The special cases in which the labor theory of value hold make obvious that, for a given technology, a higher rate of profits require a lower wage. And this wage frontier continues to hold in models of extensive rent.

8.0 Conclusion

The inclusion of natural resources, insofar as they can be modeled by extensive rent, does not seem to pose any additional issues for modern formulations of classical and Marxian political economy. It does highlight some issues that arise in models with circulating capital.

Labor values can be calculated for all produced commodities, given the technique in use. They are calculated from the marginal land that receives no rent. But suppose that a choice of technique exists. Then, an analysis at the level of prices of production must be prior to the calculation of labor values. The theory of extensive rent highlights this issue.

As Ricardo and Marx noted, prices of production are generally not proportional to labor values. They are equal in the special case, in which all industries have equal organic compositions of capital, in both models of circulating capital and of extensive rent. In the latter case, the organic composition of capital is found for agriculture from no-rent lands partially farmed.

A commodity of average organic composition is picked out in both models. Total labor values and the labor value of wages are equal to the corresponding aggregates in the system of prices of production when this average commodity is used as numeraire and is produced. These invariants, though, have to restricted to the production of the numeraire with the iron-producing process and the process on no-rent land. It is not clear to me that Marx thought his invariants held in his chapters on rent, given their location towards the end of volume 3 of Capital.

Obviously, these observations on natural resources and rent are just a start. They do seem to match what Ricardo was about in the second chapter of his Principles. The analysis of the choice of technique can be thought of, somewhat, as a critique of Ricardo.

At any rate, prices of production are well-defined in models of extensive rent. And they can be used in an analysis of the choice of technique. As usual, I present the analysis with no mention of utility maximization, preferences, or tastes.

Elsewhere

• Branko Milanovic reviews Nat Dyer's Ricardo's Dream.
• Ben Burgis explains Why Marx is back in fashion.
• Symposium on Michael Heinrich's work.
• Doug Henwood interviews John Cassidy on his new book, Capitalism and Its Critics: A Battle of Ideas in the Modern World. (Apparently there are overviews of critics I have never heard of.)
• James Surowiecki Interviews John Cassidy, also on his new book.
• Nathan Robinson interviews Richard Wolff.

Even More On Recurrence Of Truncation Without Reswitching

Figure 1: Partitions in a Parameter Space with Small a1, 1, Small a1, 3

1.0 Introduction

I have been exploring patitions of parameter spaces by fluke switch points. I always have trouble visualizing higher dimensional spaces. Sometimes, my examples are two-dimensional. If I were doing more formal mathematics, instead of numerical exploration, I would not need to care.

This post is a continuation of the example developed here, here, and here. I present two-dimensional slices of a four-dimensional space.

2.0 Perturbations of Coefficients of Production

As the result of technical change, coefficients of production vary. No variation in labor coefficients or of the output matrix are considered here so as to retain reverse labor substitution at the switch point between Beta and Delta and forward substitution of labor at the switch points between Alpha and Gamma and between Gamma and Delta. Accordingly, consider perturbations of the coefficients of production in the first row of the input matrix. These parameters define a four-dimensional space. How does the analysis of the choice of technique vary with a decrease in these coefficients? The recurrence of truncation turns out be only a transient possibility in secular time with this specific model of technical change. The reverse substitution of labor can occur without the recurrence of truncation, but is also transient. Capital reversing, also transient, occurs in one region of parameter space.

2.1 Small Amounts of Circulating Capital Needed for New Machines

The parameter space is partitioned by parameters corresponding to fluke switch points. Figure 1 shows partitions with the values of a_1,1 and a_1,3 as in this previous post. A switch point is a fluke if it is a knife edge case in which almost all perturbations of model parameters destroy its defining properties. Four of the five partitions shown correspond to a switch point on the axis for the rate of profits. With four techniques, switch points between six pairs are possible. A switch point between Alpha and Delta can only occur as an intersection of all four curves. The same goes for a switch point between Beta and Gamma. These two pairs of techniques correspond to the partition for the switch point at which all four wage curves intersect, called here a four-technique pattern.

The sequence of techniques along the wage frontier is invariant in each numbered region. Table 1 lists the cost-minimizing techniques in each region, in order of an increasing rate of profits. Consider perturbations of the coefficients of production, a_1,2 and a_1,4, that specify the inputs of corn needed for each process in which the machine is run for the second year of its life. In region 1, to the northeast, old machines require so much circulating capital to operate that it is never cost-minimizing to run them for a second year. The Alpha technique is cost-minimizing, whatever the distribution of income. An improvement in the efficiency of old machines in using circulating capital in the machine industry leads to machines being operated for two years in that industry at large rates of profits, as in region 5. A similar improvement in the efficiency of old machines in the corn industry leads to machines being operated for two years in the corn industry, as in region 2. The recurrence of truncation only occurs in region 4.

Table 1: Overview of Regions

Region	Techniques	Notes
1	Alpha	No switch point.
2	Alpha, Gamma	Lower rate of profits associated with truncation in corn industry, greater output per worker.
3	Alpha, Gamma, Delta	Lower rate of profits associated with truncation, greater output per worker.
4	Alpha, Gamma, Delta, Beta	Recurrence of truncation in corn industry.
5	Alpha, Beta	Lower rate of profits associated with truncation in machine industry, greater output per worker.
6	Gamma, Delta	Lower rate of profits associated with truncation in machine industry, greater output per worker.
7	Gamma	No switch point.
8	Beta	No switch point.
9	Delta	No switch point.
10	Delta, Beta	Lower rate of profits associated with extension of economic life in corn industry, reverse substitution of labor.
11	Gamma, Delta, Beta	Lower rate of profits associated with truncation in machine industry, extension of economic life in corn industry, reverse substitution of labor.
12	Alpha, Beta, Delta	Around the Beta vs. Delta switch point, lower rate of profits associated with truncation, smaller output per worker.
13	Alpha, Beta, Delta	Lower rate of profits associated with truncation, greater output per worker.

2.2 Large Amounts of Circulating Capital Needed for New Machines in the Corn Industry

Figure 2 shows the partitions in the parameter space at a higher level of a_1,3. The structure in Figure 1 has moved upwards and to the left. The wedge in which the recurrence of truncation appears, region 4, has widened a bit. A new partition has appeared, for a fluke switch point between Gamma and Delta on the wage axis. For parameters where the partitions between regions 2 and 3 and between regions 2 and 7 intersect, the wage frontier has fluke switch points on both the wage axis and the axis for the rate of profits. For a low enough value of a_1,4, it is no longer cost-minimizing to truncate the machine in both industries at a low rate of profits. Gamma is cost-minimizing, whatever the distribution of income, for a high enough value of a_1,2 and a low enough value of a_1,4.

Figure 2: Parameter Space with Small a1, 1, Large a1, 3

2.3 Large Amounts of Circulating Capital Needed for New Machines in the Machine Industry

Figure 3 shows the partitions in the parameter space at a higher level of a_1,1. The amount of corn needed to operate a new machine in the corn industry, a_1,3, is the same as in Figure 1. Only the partitions and regions to the right are labeled in the figure. This part of the parameter space resembles those slices examined above, with a couple of variations. To the right, the lower boundary of region 5 corresponds to a switch point on the axis for the rate of profits, not a fluke switch point in which all four wage curves intersect. The upper boundary of region 3 now is, to the right, such a four-technique pattern of switch points.

Figure 3: Parameter Space with Large a1, 1, Small a1, 3

Figure 4 is an enlargement of the lower left of this slice of the parameter space. Three new points that are a fluke in multiple ways have appeared. One point corresponds to the intersection of the four wage curves on the wage axis. This point is the intersection of five partitions in the parameter space, just as the point in parameter space corresponding to the intersection of the four wage curves on the axis for the rate of profits. The region in which the recurrence of truncation occurs, region 4, has appeared. The reverse substitution of labor also occurs in regions 10 and 11. Two points in the part of the parameter space depicted correspond to fluke switch points simultaneously lying on both axes.

Figure 4: Parameter Space with Large a1, 1, Small a1, 3 (Enlarged)

Figure 5 enlarges this slice of the parameter space in the middle of Figure 3. The sequence of cost-minimizing techniques, in order of an increasing rate of profits, is Alpha, Beta, and Delta in both region 12 and region 13. The boundary between regions 12 and 13 corresponds to an intersection of the Beta and Delta wage curves on the waxis but not on the frontier. In region 12, the wage curves for Beta and Delta intersect twice in the first quadrant. The second intersection is on the frontier. It is a switch point exhibiting capital-reversing. Around the switch point between Beta and Delta a lower rate of profits is associated with a decreased capital-intensity and decreased net output per worker in the economy as a whole.

Figure 5: Parameter Space with Large a1, 1, Small a1, 3 (Another Enlargement)

3.0 Conclusion

The exploration of partitions in parameter space supports an analysis of technical change in which the circulating capital needed to operate a machine decreases. A trajectory from the upper right to the lower left in the figures and from Figures 2 or 3 to Figure 1 represent this specific form of technical change.

This analysis illustrates how perturbing coefficients of production affects the analysis of the choice of technique. Parameter spaces are partitioned by fluke switch points into regions in which the variation of the cost-minimizing technique with distribution is itself invariant. Some of the structures formed by intersections of partitions are not specific to the example, or even to models of fixed capital. For example, fluke switch points can arise in many models simultaneously on both the wage axis and the axis for the rate of profits. Perturbations of parameters around such a point eliminates or creates the possibility for certain techniques to be cost-minimizing at extreme ranges for the rate of profits.

Likewise, perturbation analysis supports the analysis of specific forms of technical progress. By contrast with fluke switch points, the recurrence of truncation, the reverse substitution of labor, and capital-reversing are not flukes. The reverse substitution of labor occurs in regions in which the recurrence of truncation does not occur. In this sense, the reverse substitution of labor is more general in this example. Capital-reversing arises in a region in which neither of these other 'perverse' phenomena occur. Configurations of parameters in which these phenomena arise, however, define only a few regions of the parameter space. This observation seems capable of generalization to other 'perverse' phenomena in other examples. Other forms of technical progress can be explored.

On Confusion About Gödel's Theorem, Including In Austrian Economics

1.0 Introduction

Gödel's theorems are often invoked in non-mathematical contexts, sometimes in a very imprecise fashion (Franzen 2005, Raatikaine 2007, Jaimungal). You should be skeptical of what many say about Gödel, including of what I say.

In this post, I look at two examples. One is of a confused economist of the Austrian school. The other is of Wittgenstein, who has been ably defended on this point.

2.0 A Claim About the Economic Calculation Problem

Nguyen (2024) says that somehow Gödel's theorem supports the claim that centralized economic planning is, in principle, impossible. Nguyen is sympathetic to the claim that our minds (brains?) transcend the capabilities of all formal systems. I find this claim dubious, but I have not read Penrose. For purposes of argument, Nguyen assumes, through most of this paper, that the central planner has the information von Mises grants them. Nguyen deliberately puts aside Hayek's concerns about non-articulated, distributed, tacit knowledge.

I have demonstrated that von Mises' argument is invalid. I find I am not original. Cockshott (2010) has done the same. Both of us put forth a linear programming formulation that does not require prices of intermediate goods as part of the data. We differ in our specifications of the planner's objective function. Neither of us are echoing Lange and Lerner's formulations of general equilibrium. As such, I do not see any issues are raised for us by the non-computability of utility functions, preference relations, or general equilibria in which ratios of marginal utilities enter the system of equations.

A polynomial-time algorithm exists for solving linear programs. Linear programs are not undecidable. So Gödel's theorem and issues arising from Turing's work do not seem to have any purchase here.

Furthermore, in practice, only a finite number of numbers would be used in drawing up plans. Real numbers would be approximated, if you can call it that, by IEEE Std. 754. This standard defines floating-point numbers, in single and double precision formats. Your computer does not only fail to represent the full range of real numbers. It also only represents a finite number of integers in words, typically of 32 or 64 bits.

I do not mean to suggest that many practical issues do not arise with central planning Nor am I advocating such. I continue to maintain that von Mises' argument is invalid. And I find Nguyen's paper to be one of those mystical, imprecise invocations of Gödel.

3.0 A Notorious Paragraph from Wittgenstein

Here is Wittgenstein from notes not published until after his death:

"8. I imagine someone asking my advice; he says: 'I have constructed a proposition (I will use "P" to designate it) in Russell's symbolism, and by means of certain definitions and transformations it can be so interpreted that it says: "P is not provable in Russell's system". Must I not say tha this proposition on the one hand is true, and on the other hand unprovable? For suppose it were false; then it is true that it is provable. that surely cannot be! And if it is proved, then it is proved that it is not provable. Thus it can only be true, but unprovable.'

Just as we ask, '"Provable" in what system?,' so we must also ask, '"True" in what system?' 'True in Russell's system' means, as was said, proved in Russell's system, and 'false in Russell's system' means the opposite has been proved in Russell's system. - Now what does your 'suppose it is false' mean? In the Russell sense it means, 'suppose the opposite is proved in Russell's system'; if that is your assumption you will now presumably give up the interpretation that it is unprovable. And by 'this interpretation' I understand the translation into this English sentence. — If you assume that the proposition is provable in Russell's system, that means it is true in the Russell sense, and the interpretation 'P is not provable' again has to be given up. If you assume that the proposition is true in the Russell sense, the same thing follows. Further: if the proposition is supposed to be false in some other than the Russell sense, then it does not contradict this for it to be proved in Russell's system. (What is called 'losing' in chess may constitute winning in another game.)" -- Wittgenstein (1978: Part I, Appendix III)

Wittgenstein seems to equate provability in PM with being true in PM. Is not part of Gödel's point to separate these concepts? Furthermore, Wittgenstein seems to be writing only about the heuristic argument given at the start of Gödel's paper. Do his remarks make sense of Gödel numbering, (primitive) recursive functions, and so on?

As I understand it, Gödel's proof of his incompleteness theorem is a conventional proof. The proof is an argument to convince you that a sequence of statements exists that follow one another, by conventional deduction rules. And the conclusion is a theorem about natural numbers.

Gödel's proof is about the syntactic manipulation of strings of symbols by formal rules. Wittgenstein questions interpretations, I guess, of Godel numbers as the statements or sequence of statements that map into them. Floyd and Putnam (2000) justify Wittgenstein in a way that draws on the distinction between consistency and ω-consistency.

J. B. Rosser replaced ω-consistency with consistency in Gödel's proof. A theory is ω-inconsistent if one can show, for some proposition p:

• not p(0), not p(1), not p(2), ..., and
• There exists x such that p(x).

I guess that a non-standard model can be ω-inconsistent, where x is not a natural number, somehow. Suppose PM is ω-inconsistent. And suppose the negation of the Gödel sentence 'P' is true. Then the Godel number for the proof of this proposition could be a non-standard natural number. Wittgenstein writes about interpretations, but does not mention ω-consistency. I suppose he could have known about the Löwenheim-Skolem theorem.

I am sympathetic to being suspicious of English-language interpretations of syntactical manipulations. I am not so sympathetic to how lightly Wittgenstein treats possible contradictions in mathematics. I am sympathetic to the idea that mathematical logic, set theory, and model theory, for example, just provides more maths. One does not need this math to justify what humans have been doing for millennia. Do mathematical propositions have meaning before proofs are found? As I understand Wittgenstein, he says not. Proofs draw connections and give the proved proposition a meaning.

But I am also aware that even if I can echo out some propositions from these fields, I am no expert in them.

References

• Cockshott, W. Paul. 2010. Von Mises, Kantorovich and in-natura calculation. European Journal of Economics and Economic Policies 7(1): 167-199.
• Dawson, Ryan. 2015. Leaving Mathematics As It Is: Wittgenstein's Later Philosophy of Mathematics. University of East Anglica. PhD Thesis.
• Floyd, Juliet and Hilary Putnam. 2000. A note on Wittgenstein’s 'notorious paragraph' about the Gödel theorem. Journal of Philosophy. 97(11): 624-632.
• Franzen, Torkel. 2005. Gödel's Theorem: An Incomplete Guide to Its Use and Abuse. A. K. Peters.
• Jaimungal, Curt. Theory of Everything: a podcast episode.
• Nguyen, Hai-Trieu. 2024. The incompleteness of central planning. Quarterly Journal of Austrian Economics. 27(4).
• Raatikaine, Panu. 2007. Review of Franzen. Notices of the AMS. 54(3): 380-383.
• Standford Encylopedia of Philosophy, Wittgenstein's philosophy of mathematics.
• Wittgenstein, Ludwig. 1978. Remarks on the Foundation of Mathematics, revised edition. MIT Press.