I wrote this piece for the National Institute for Economic and Social Research's

*Rebuilding Macroeconomics*project.
There are
three types of mathematicians: those that can count and those that can’t. This
aphorism challenges the public perception of mathematics as being concerned
with calculation and is liked by mathematicians because it enables them to
highlight what mathematics is concerned with, which is identifying and
describing relationships between objects.

A more
sophisticated misunderstanding relates to the way mathematics is conducted. The
error originates in how mathematicians present their work, as starting with
definitions and assumptions from which ever more complex theorems are deduced.
This is the convention that Euclid established in his

*Elements of Geometry*and led Kant to believe that*synthetic**a priori*knowledge was possible. Euclid actually started with Pythagoras’ Theorem, and all the other geometric ‘rules’ that had emerged out of practice, and broke them into their constituent parts until he identified the elements of geometry. It was only having completed this analysis did he then reconstruct geometry in a systematic way in*The Elements*. Today the consensus within mathematics is that the discipline is*analytic*, from observations, not*synthetic,*outside of mathematics there persists a belief in the power of pure deductive, synthetic*a priori*reasoning.
Physical
sciences are in tune with what mathematicians do. This is exemplified by Newton
who gathered observations on the planets and invented calculus to interpret the
data. On this basis he concluded that momentum was being conserved and deduced
the gravitational law. The key idea originating with Newton is that momentum is
an invariant in a dynamic system. This is understood most clearly when
presented using calculus, the mathematics Newton invented. Since Newton, all
significant advances in physics have been associated with the identification of
an invariant (momentum, energy, increase in entropy, speed of light) and inventing
clear and succinct ways of describing objects (mathematics) that re-presents
nature based on an invariant.

Finance has
developed a mathematical theory in the Fundamental Theorem of Asset Pricing
that has the same status in mathematical finance as Newton’s Laws have in
classical physics. The central principle, analogous to the conservation of momentum,
is that of ‘no-arbitrage’. The Fundamental Theorem of Asset Pricing states that
if an asset is priced on the principle of no-arbitrage then there is a
reciprocal relationship in the exchange. There are at least two ways of
understanding this principle. It is a version of Euclid’s ‘First Common Notion’:
if

*A=B*and*C=B*, then*A=C*. Money takes the role of “*B*” and arbitrates the value of*A*relative to*C*. Alternatively, it is a version of the scholastic argument that a riskless profit is a shameful gain (*turpe lucrum*).
The no-arbitrage
principle is justified through Ramsey’s ‘Dutch Book Argument’ that requires
markets are mediated by jobbers (market-makers or dealers in the US) rather
than brokers. When a jobber quotes a price, they do not know whether the
counter-party is looking to buy or sell at the price. The jobber will quote a
price at which they will buy and a higher price at which they will sell. They
signify confidence in their quote by having a narrow difference between the prices.
If a jobber quotes a price that another trader believes is wrong, the trader
will take the quote, immediately moving the market. These jobber-mediated
markets are, therefore, essentially discursive. Jobbers are engaged in making
assertions as to prices, which are challenged when others take the quote; this
is ‘market making’. If the market agrees that a jobber has correctly priced the
asset, no trading will take place - silence is consent - and the market
dissolves.

Jobbers do
not hold assets and prefer trading in financial contracts rather than hold
physical assets, they have no commitment to the assets they trade and identify
themselves as taking long and short positions rather than buying or selling.
While they lack commitment to assets, jobbers must be sincere in their
statements, they must believe the quote is right. This means, that in the face
of radical uncertainty, a jobber’s price quote is reliable, it can be trusted.

The
significance of reciprocity in markets rests on the no-arbitrage principle that
can only be justified if exchange is being conducted by jobbers, who will buy
and sell at the quoted prices. Markets in economics tend to be based on brokers
who bring property owners, one a buyer, one a seller, together. The focus on
broker-mediated markets rather than jobber-mediated markets means that the
importance of reciprocity in exchange is obscured. The different emphasis is rooted
in financial markets being concerned with uncertain futures whereas economic
markets are concerned with immediate scarcities.

In modern
business, if a manufacturer can sell a product at an enormous profit, creating
an arbitrage, they are succeeding. Economic theory argues that in the presence
of these excess profits, competitors can come in and the price of the product
will fall. This appears to be no different to the situation in a
jobber-mediated market: jobbers will bid (buy) at the cost of production and offer
(sell) at that cost plus a risk premium, just as manufacturers will do in a
competitive broker-mediated market. However, while the ultimate point might be
the same for jobber and broker mediated markets, the routes to the point are
different. For jobbers, no-arbitrage, and hence reciprocity, are iron laws that
must not be breached, ever. In broker-mediated markets, arbitrages are
transitory and the ideal is to capture them before they disappear; it is a
virtue to break the principle of reciprocity. Prices at which exchange takes
place in jobber-mediated markets are always disputed prices, but sincere; in
broker-mediated markets, prices are always accepted, if not fair.

Consider some
thought experiments. If a manufacturer, making arbitrage profits, was obliged
to buy identical goods, manufactured by others, at the prices they themselves quoted,
would they quote the same price? If a slum-landlord had to live in the accommodation
they rented, would they rent inferior quality accommodation? Public services
are often expensive because they are of a quality that the providers would like
to receive. These examples highlight the ethical nature of dual-quoting, it
imposes the categorical imperative: do unto others as you would have them do
unto you.

Financial
instability has long been blamed on jobbers, who trade ‘paper’ and lack
commitment to material assets, they are 'disinterested'. However, bubbles are a consequence of property
owners ‘ramping’ assets and selling them above their intrinsic value. The
failure of Long Term Capital Management in 1997 was precipitated by an apparent
arbitrage, in the ‘asset swap’ strategy involving rock-solid US government
debt, being an illusion. The Credit Crisis was a result of investment banks
believing they could construct mortgage backed securities (MBS), out of ‘real’
assets, for less than their worth, not realising the inherent risks because
they believed in arbitrages. Investment banks have been fined for selling MBS
above their internally recognised value; they were being profit maximisers but
insincere. There is evidence that the ‘Bitcoin’ bubble of December 2017 was a
consequence of it being easy to buy Bitcoin, but difficult to sell; something
not possible in a jobber-mediated market. These are all situations where
financial instability originates in a belief that the no-arbitrage principle
could be ignored or that prices could be insincere, and so it was possible to earn
risk-less profits.

Recognising
that the no-arbitrage principle is analogous to Euclid’s First Common Notion
means that arbitrageurs should be regarded in the same way as promoters’
perpetual-motion-machines are: mis-guided cranks. It also emphasises that
exchange should be reciprocal, it should not involve profiting at another’s
expense. Mathematics only works on the basis of Euclid’s First Common Notion;
markets only work well on the basis of reciprocity.

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