Vectors and Values

In the first edition of Capital, Vol 1, Ch 1, Marx mentions that:

 Both forms, the relative value-form of the one commodity, equivalent form of the other, are forms of exchange-value. Both are actually only vectors – determinations conditioned reciprocally by each other – of the same relative value-expression, but divided like poles between the two commodity-extremes which have been set equal. (Translation by Albert Dragstedt

I have emphasized the word “vectors” because I had not noticed it before, but I had previously been struck by the necessity to translate the abstruse “Hegelian” discussion of value forms into the modern mathematical language of equivalence relations. This should go together with the modern treatment of proportions and differences as vectors to construct the rationals and the integers from “natural” (positive) magnitudes. Also connect with modern duality theory, convexity, optimum control, separating hyperplanes etc etc and refer back to Marx’s letter to Dr Kugelmann of July 11 1868 explaining that value is simply the particular social form taken by the necessary proportionalities between different kinds of labor.

Note: 2018-10-12

There is another translation:

doi = {10.1177/030981687800400111},
title = {The Value-Form},
author = {Marx, K.},
publisher = {Sage Publications},
journal = {Capital & Class},
issnp = {0309-8168},
issne = {2041-0980},
year = {1978},
month = {01},
day = {01},
volume = {2},
issue = {1},
page = {134–150},
url = {},

This translation does not use the term “vector”. Introduction has some useful discussion of other issues and says translation by Dragstedt had various flaws. No specific reference to translation of “vectors”.

doi = {10.1177/030981687800400110},
title = {The Value-Forum},
author = {Marx, K.},
publisher = {Sage Publications},
journal = {Capital & Class},
issnp = {0309-8168},
issne = {2041-0980},
year = {1978},
month = {01},
day = {01},
volume = {2},
issue = {1},
page = {130–133},
url = {},

Use of the term “vector” by Marx seemed anachronistic. Quick look at Hegel’s Philosophy of Nature Vol 1 did not shed any light. (But did have stuff about points and lines that may shed light on Marx’s discussion of swerves in his doctoral thesis on quantum mechanics).

Must check for reference in other versions and manuscripts. (Also related to “moments”, with inseparable magnitude and direction cf inseparale length and force in lever torque!)

Looked up “vectors” and came across “Giusto Bellavitis” credited with first exposition of what are now called vectors via “equipollence” (equivalence relation defining an affine space) in 1835. Apparently he also corresponded with Walras on this precise issue!

Here is a modern explanation, showing its superiority to the usual clumsiness.

Also mentioned in German biography of Grassmann. Must come back to this when studying Grassmann/Lawvere/univalent modal homotopy type theory!

Have downloaded “History of Vector Analysis” by Michael J. Crowe. Looks like Marx could have picked up from Leibniz/Ancient Greeks/Complex numbers (all of which he studied closely). Worth reading history carefully and perhaps tracking down in Marx.

Also downloaded several volumes on Grassmann!

Including German language biography by Hans-Joachim Petsche, which linked Marx and Bellavitis and Grassmann in a random google search. See notes 229 to 242 on p110. Might be worth getting rough translation (eg google) if not found in english volumes. (Since found similar in english version of biography by same author – did not look significant but check later when actually reading stuff on Grassmann).

Have previously downloaded “Polarity and analogy; two types of argumentation in early Greek thought” by G.E.R. Lloyd.

Also may relate to Douglas Hofstadter on “Fluid concepts and creative analogies” re concepts/categories/analogies and Freyd and Scedrov “Categories, Allegories”.

After all, panta rhei and everything is connected to everything else.

Meanwhile hopefully this gets it out of my head for a while.

Except reminded to include Galileo’s verbal version of kinematics that is vastly easier to understand as simple algebra than verbally via Bill Kerr at Strange Times. See “Changing Minds: Computers, Learning, and Literacy”  by Andrea A. diSessa (chapter 1, especially p13):

THEOREM 1 If a moving particle, carried uniformly at constant speed, traverses two distances, then the time intervals required are to each other in the ratio of these distances.

THEOREM 2 If a moving particle traverses two distances in equal intervals of time, these distances will bear to each other the same ratio as their speeds. And conversely, if the distances are as the speeds, then the times are equal.

THEOREM 3 In the case of unequal speeds, the time intervals required to traverse a given space are to each other inversely as the speeds.

THEOREM 4 If two particles are carried with uniform motion, but each with a different speed, then the distances covered by them during unequal intervals of time bear to each other the compound ratio of the speeds and time intervals.

THEOREM 5 If two particles are moved at a uniform rate, but with unequal speeds, through unequal distances, then the ratio of the time intervals occupied will be the products of the distances by the inverse ratio of the speeds.

THEOREM 6 If two particles are carried at a uniform rate, the ratio of their speeds will be the product of the ratio of the distances traversed by the inverse ratio of the time intervals occupied.

A modern reader (after struggling past the language of ratios and inverse ratios) must surely get the impression that here there is much ado about very little. It seems like a pretentious and grandly overdone set of variations on the theme of “distance equals rate times time.” To make matters worse, the proofs of these theorems given by Galileo are hardly trivial, averaging almost a page of text. The first proof, indeed, is difficult enough that it took me about a half-dozen readings before I understood how it worked. (See the boxed text.)

In fact this is a set of variations on distance equals rate times time. Allow me to make this abundantly clear. Each of these theorems is about two motions, so we can write “distance equals rate times time” for each. Subscripts specify which motion the distance (d), rate (r), and time interval (t) belong to.

d_1 = r_1 \cdot t_1
d_2 = r_2 \cdot t_2

In these terms, we can state and prove each of Galileo’s theorems. Because Galileo uses ratios, first we divide equals by equals (the left and right sides of the equations above, respectively) and achieve:

\frac{d_1}{d_2}= \frac{r_1 \cdot t_1}{r_2 \cdot t_2}

THEOREM 1 In the case r1 = r2, the r terms cancel, leaving d1/d2 = t1/t2.

THEOREM 2 In the case t1 = t2, the t terms cancel, leaving d1/d2 = r1/r2. Conversely, if d1/d2 = r1/r2 then t1/t2 = 1 or t1 =t2.

THEOREM 3 In the case of d1 = d2, the d terms cancel, leaving (r1/r2)(t1/t2) = 1, or t1/t2 = r2/r1.

THEOREM 4 This is precisely our little ratio lemma, d1/d2 = (r1/r2)(t1/t2).

THEOREM 5 Solve the equation above for t1/t2; t1/t2 = (d1/d2)(r2/r1).

THEOREM 6 Solve for r1/r2; r1/r2 = (d1/d2)(t2/t1).

For direct contrast, I reproduce Galileo’s proof of theorem 1, which is one-sixth of the job we did with algebra, in box 1.

So now we’ve redone a significant piece of work by one of the great geniuses of Western science, with amazing ease. Solving problems is always easier after the first time around, but the difference here is almost mindboggling. What we did would constitute only an exercise for a ninth-grade mathematics student.

That, in fact, is the key. Galileo never had ninth-grade mathematics; he didn’t know algebra! There is not a single “=” in all of Galileo’s writing.

The fault is not with Galileo or with the education provided by his parents or with the schooling of the times. Algebra simply did not exist at that time. To be more precise, although solving for unknowns that participated in given relations with other numbers had been practiced for at least half a millennium, the modern notational system that allows writing equations as we know themand also the easy manipulations to solve themdid not exist. Fifty years after Galileo’s main work, René Descartes (1596-1650) would have a really good start on modern algebra. Later, by the end of the seventeenth century, algebra had stabilized to roughly the modern notation and manipulative practices, although it would be the twentieth century before algebra became a part of widespread technical literacy.

Theorem 1 and Galileo’s axioms refer to a particle being “carried”. Origins of “vector” is the Latin for “carrier” (also used for disease carriers like mosquitos and for gene carriers). Check whether Galileo’s original Latin/Italian uses the word vector. (Have downloaded original. Very quick look suggests he did not. Check again when actually reviewing the long boxed proofs (in english).

Consider doing parallel treatment of weight as defined by Euclid in terms of “heaviness” etc with axiomatization of both measurement and the concept itself as in “Foundations of measurement. Vol. 1. Additive and polynomial representations” by David H Krantz; R D Luce; Patrick Suppes; A Tvesky. Find simpler logical treatment of additive magnitudes by Suppes and the well known theorem about sub-magmas of reals (when Archimedean) mentioned in Birkhoff on Lattices. Also Tarski.

Could combine with historical treatment – weighing grain and gold with scales after long epochs of occasional gifts and exchanges with neighbouring communities etc etc. Better to follow Marx’s recommendation and start with “Contribution” before chapter 1 of Capital. Highlight role of arbitrage and merchants along with money.

Then discover “intrinsic” property is mass, not weight. Caricature “transformation problem” considering density when immersed in fluids more dense than air. Further frictional, viscous, aerodynamic, hydrodynamic and then relativistic complications then curvature of spacetime and whatever comes next when measuring dynamically instead of purely hypothetical “static equilibrium”.

Saw a development of money from barter expressed in modern notation. Could take a lot further with markov chains etc. Give actual proof that circulation expels a single money commodity. The extremely efficient algorithms for disjoint set data structure could illustrate how spontaneous arbitrage in exchange rapidly produces relatively stable partitioning into equivalents and selection of a money commodity as universal equivalent.

Also show full development of commodities naturally follows rather than precedes capital and wage labour where inputs and outputs both commodities and purpose is capital increasing itself.

But experience has shown that Marx’s method of presentation simply did not work. As Lenin said, decades later, none of the Second International understood it. (“Understanding” was a pretty low bar, need more for actually completing and developing it.)

Try opposite sequence. Start describing the appearance, including modern aggregate economic and related statistics and their changing distributions, history of regular crises, current situation. Explain current theories that don’t work. Need to get behind the appearance and extract essence of the dynamic. Comparison with other sciences (solar system etc). Then refer to established categories of value, money, affordance, demand, supply, input/output, activity, credit, yield curve etc.

Then offer (improved) Maksakovsky “pure” toy model as dynamic theory based on development of contradictions from reproduction schemes resolved cyclically in a spiral that both develops the level of technology and intensifies the contradictions of capitalism that prepare the way for revolutionary transition including concentration, centralization, globalization etc. (Ties in very well with the good bits of Stalinist “New Philosophy” in Leningrad textbook).

On the logical/mathematical side, hopefully some additional axioms similar to Galileo/Suppes/Tarski could bring out lattice normed bounded complete vector space of commodity bundles which are sufficient basis for lots of functional analysis and mathematial economics (eg to explain values “duality” as in linear programming etc (Kantorovich Stalin and Nobel prizes). Hopefully Grassmann’s Geometric Algebra (including Geometric Calculus) brings out “moments” and curved space better. See “New Foundations in Mathematics” by Sobczyck.

See also Kutateladze et al developing Kantorovich spaces etc from reals via forcing invertible infinitesimals with Boolean valued models. Hopefully can do similar with the more classical nullsquare infinitesimals in toposes.

See also Aliprantis for lots more duality theory with cones, positive operators etc in mainstream “General Equilibrium”.

Real dynamics of stochastic disequilibrium as in Marx and Maksakovsky much harder.

Delays central to Maksakovsky’s cycle. Differential inclusions, not equations (as in optimal control, differential games etc). “Variational Analysis” by Rockafellar and Wets based on set valued analysis. Path dependent stochastics as in “A Modern Theory of Random Variation” by Patrick Muldowney.

So studying “path dependent stochastic delay partial differential inclusion games” aka “chaotic evolution of recursively self-organizing complex systems” aka materialist dialectics. More likely to find useful stuff in mathematical biology literature than in economics literature.

Note: 2018-10-11

Also checkout Krause.

Could one develop vectors from partial magma of sporadic exchange relations unified or potentially unified by money commodity?

Cancellative unital medial partial magma gives positive cone of commutative group?


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