19 August 2014

On the common belief that 1 + 1 = 2

School children are taught at an early age that 1 apple plus 1 apple equals 2 apples. They go on to learn that 1 orange plus 1 orange equals 2 oranges. Sometimes they painfully discover on their own that 1 caning plus 1 caning equals 2 canings. Eventually they conclude that 1 + 1 = 2, irrespective of what is placed immediately after the 1s and 2 or even if there is nothing after the 1s and 2. The belief that 1 + 1 = 2 is a universal truth is widespread, even among mathematicians. But is it true?

In primary school students are posed with problems of the following kind. If two apples cost Rs 20, how much would four apples cost? And they are taught that it can be solved in this fashion.

First you arrange the data as below:
2      20
4      ?

Then, if you cross multiply you find that ? (or what is in later years called the unknown variable) is equal to 40.

In primary school this seems to be a universal truth. But in secondary school the student comes in for a rude shock. He finds that there are problems that cannot be solved in this manner and in fact yield the wrong answer if he attempts to solve them this way.

Thus, if 2 workers take 20 hours to build a wall, then it is incorrect to conclude that 4 workers will take 40 hours to build the same wall. Before one can solve such a problem, one needs to decide in advance whether it is one of direct or inverse proportion.

A similar argument applies to the equation 1 + 1 = 2. Imagine that I eat one apple between 8 am and 9 am and one apple between 9 am and 10 am. How many apples do I eat between 8 am and 10 am. An operational way of arriving at an answer would be to place 1 on the number line for the first apple and then move 1 unit to the right for the second apple to find that between 8 am and 10 am I have eaten 2 apples.

Imagine, now, that I travel at a speed of 20 km/hr between 8 am and 9 am and at a speed of 40 km/hr between 9 am and 10 am. I cannot use the same operational method to conclude that between 8 am and 10 am I have travelled at a speed of 60 km/hr. Distance objects are amenable to the rule that 1 + 1 = 2 but velocity objects are not. We instinctively apply the 1 + 1 = 2 rule to apples and distances but not to velocities and think no more of it.

At the start of the 20th century A.N. Whitehead and Bertrand Russell set out to prove, among other things, that 1 = 1 = 2, from even more fundamental notions. The result was Principia Mathematica which was published in three volumes in 1910, 1912 and 1913. Ludwig Wittgenstein raised an immediate objection that the book used a circular argument in that by using sets it was using objects that were known to obey the 1 + 1 = 2 rule. But the prestige of Whitehead and Russell was such that the book was still acclaimed widely as having achieved a breakthrough.

Some years earlier Gottlob Frege had travelled a similar path. Just when his magnum opus The Fundamental Laws of Arithmetic was dusted and ready for publication he received a letter from Russell asking whether the set of all sets which are not members of themselves was a member of itself. Frege was gracious enough to conclude his second volume with this acknowledgement: "A scientist can hardly encounter anything more undesirable than to have the foundation collapse just as the work is finished. I was put in this position by a letter from Mr Bertrand Russell when the work was almost through the press."

Russell was to suffer a similar fate himself when Kurt Godel in 1931 published a paper titled "On Formally Undecidable Propositions in Principia Mathematica and Related Systems". Wikipedia paraphrases the paper thus: "Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory." More simply, you cannot prove that 1 + 1 = 2.

But formal proofs apart, there is another reason for doubting the universal truth of 1 + 1 = 2. Primary school mathematics is divided into two compartments: arithmetic and geometry. For centuries it was believed that the only geometry possible was Euclidean geometry in which parallel lines meet at infinity and the sum of the measures of the three angles of a triangle is 180 degrees. At the beginnng of the 19th century Gauss discovered the principles of non-Euclidean geometry but the idea was so outlandish that he did not publish. It was left to the Hungarian mathematician Janos Bolyai and the Russian mathematician Nikolai Ivanovich Lobachevsky to lay out the basic principles around 1830.

Even so it was not thought to be anything more than an academic exercise. Only after Einstein published the General Theory of Relativity in 1916 was it realised that the world was basically non-Euclidean and that space was not flat but curved and that the curvature varied from point to point.

If school geometry was only one of many possibilities then from symmetry considerations one should conclude that the school arithmetic in which 1 + 1 = 2 is also only one of many possible arithmetics. In fairness to Whitehead and Russell it must be pointed out that their work was published before Einstein's General Theory.

Thus one is led to the conclusion that if one cannot prove that 1 + 1 = 2 it is because 1 + 1 is not equal to 2. In deciding to apply the rule to distances but not velocities we demonstrate that we know this. But because so many of the objects of our everyday experience do follow the 1 + 1 = 2 rule we conclude that it is universally true. But this is no more than a reflection of the fact that as human beings we tend to reason inductively, accept those facts which fit our beliefs and thus reinforce them, and reject those facts which do not fit our beliefs.

What we can say at best therefore is that 1 + 1 is sometimes equal to 2 though perhaps we can even go so far as to say that 1 + 1 is often equal to 2.

Category: Mathematics


13 May 2014

The mathematical equivalence of Keynesianism and monetarism

It has happened often in physics that a single phenomenon is explained, or a single puzzle resolved, by two theories that seem at first sight to be completely divergent but are later shown to be equivalent. Examples that spring to mind are Heisenberg's matrix mechanics and Schrodinger's wave mechanics or the quantum electrodynamic theories of Tomonaga, Schwinger and Feynman. In macroeconomics, the second half of the 20th century was dominated by the dispute between Keynesianism and monetarism, especially their divergent explanations of recessions, a dispute that continues to this day. This paper demonstrates that the conflict hinges on a simple dimensional misinterpretation of one of the variables in the quantity theory of money. At their heart, the two theories are equivalent.

Read the entire paper: The mathematical equivalence of Keynesianism and monetarism

Category: Economics


09 May 2014

A few thoughts on Thomas Piketty

Primitive tribes must be the most equal of societies. Few of us, though, aspire to that kind of equality.

The thought occurred on reading about (note, not reading) Thomas Piketty's Capital in the 21st century that seems set to be the best-selling economics text of recent times. Piketty, who has in some quarters been hailed as a latter-day Marx, notes that income inequality in the US remained stable from 1910 to 1920, rose from 1920 to 1929, fell steeply after the Great Crash of October 1929 until the end of the war, remained stable until around 1980, and then rose steadily again, until in 2007 it rose above the level of 1928. A graph can be seen on Piketty's web site. To set right what he sees as a dire situation, possibly to prevent a capture of western governments by its poverty-stricken masses, Piketty suggests a general wealth tax and a top income tax rate of 80%.

Piketty seems to think that greater equality is something much to be desired. To test this I searched on the net for inequality measures for the Soviet Union to compare with the US. And I found some interesting figures in a paper Income Distribution in the USSR in the 1980s by Michael V. Alexeev and Clifford G. Gaddy. For the US some comparable figures can be found on the Federal Reserve Bank of St Louis web site.

The table below compares the two:
Gini Coefficients for the USSR and the US
YearUSSRUSA
19800.2900.403
19850.2840.419
19880.2900.426
19890.2750.431
19900.2810.428

Readers may recall that the only revolution that happened was in the USSR, not in the US.

Poring over English factory inspector reports in the sixties and seventies of the 19th century Marx reached the conclusion that the overthrow of capitalism was imminent. If nothing else, Marx's prognostications should serve as a warning that one must not use short-term data to jump to eternal conclusions. In the graph the current trend of rising inequality dates from around 1980. Is there any other variable that could explain this as well as the shifts in inequality mentioned earlier: stable from 1910 to 1920, a rise from 1920 to 1929, a fall thereafter until 1945, stable until 1980, and a rise thereafter?

It is illuminating to look at the following graph of US long term interest rates.

It is taken from The real rate of interest from 1800-1990: A study of the US and UK by Jeremy J. Siegel. The graph of inequality on Piketty's site and the interest rate graph here follow a similar trajectory. The period from 1910 to 1920 is a period of rising rates and stable inequality. Thereafter the interest rate falls and inequality grows. Similarly the period from 1980 is a period of rising inequality, and interest rates begin to fall from around that date. The Depression years were an exception. So were the war years but then that was a period of wage and price controls.

One cannot help but feel that low interest rates help push up asset prices and thus boost those who earn a substantial part of their income from financial assets. Now it so happens that the people who complain about rising inequality, Paul Krugman to take one example, are also the ones clamouring loudest for keeping interest rates low. Talk about the law of unintended consequences.

Category: Economics


02 May 2014

The dimensions of the velocity of money

A mathematical equation that correctly describes a physical relationship between quantities is dimensionally homogeneous. However, the converse is not true. An equation that is dimensionally homogeneous does not imply the existence of a physical relationship between the quantities in that equation. The dimensions of the velocity of money are generally taken to be t-1. In what follows we examine this assumption and show that it is physically impossible. We then show what the correct dimensions of velocity are and arrive at the surprising inference that at their core Keynesianism and monetarism amount to the same thing.

Read more

Category: Economics


02 April 2014

The velocity of money is a function of interest rates

The velocity of money has a way of frustrating the best-laid plans of central banks. For instance, when the Fed wants to squeeze the growth of money it often finds that velocity increases sharply, so that the same amount of money does much more work than before, thus rendering the constriction of money ineffective. Similarly, when the Fed wants to expand the amount of money, it often finds that the velocity of money falls sharply.

The unpredictability of money velocity was a key factor in hastening the demise of monetarism in the eighties. Economics, the textbook by Paul Samuelson and William Nordhaus, said in the 2005 edition: "As the velocity of money became increasingly unstable, the Federal Reserve gradually stopped using it as a guide for monetary policy... Indeed, in 1999, the minutes of the Federal Open Market Committee contain not a single mention of the term 'velocity' to describe the state of the economy or to explain the reasons for the committee's short-run policy actions."

Economists since Milton Friedman have sought to relate velocity to a number of factors and have been unsuccessful. The graph below will therefore come as an utter surprise to many. It plots the velocity of money against Moody's Seasoned AAA Corporate Bond Yield. For the entire period of five decades, 1960 to 2011 it shows that velocity runs a course exactly parallel to corporate bond yield. No one can be more surprised than I am. The velocity of money graph occurs in my book "The General Theory of Money" published in May 2012 but until now I had never thought of connecting it with interest rates if only because all the papers I had read never spoke of any relation between the two.

The monetary aggregate used in the graph is what I have called Corrected Money Supply in my book. A brief explanation is in order. Assume that you receive a salary of $1000 at the start of every month into your demand deposit. During the course of the month you spend 95% of this and save 5%. So the demand deposit would start at $1000 and run down to $50. However, you choose to keep a little extra in your demand deposit to allow for exigencies, say $500, roughly the value of your demand deposit at the middle of the month. If you add all the funds in all demand deposits in the economy there would thus be a certain amount of money that is never spent. Corrected Money Supply is M1 reduced by this money that is not a medium of exchange but is held purely with a precautionary motive.

One economist who came close to identifying the relation between money velocity and interest rates was John Tatom. In a 1983 paper called "Was the 1982 velocity decline unusual?" published by the Federal Reserve Bank of St Louis he observed that during numerous recessions after 1947 the velocity of money fell. He was, however, puzzled by the fact that in the 1970 and 1973-75 recessions the velocity rose. After some analysis he concluded: "Explanations that focus on declining interest rates also do not match up well with the recent pattern of velocity declines. In the first quarter of 1982, corporate Aaa bond yields averaged 15.01 percent and had risen from 14,62 percent one quarter earlier or 14.92 percent two quarters earlier. During the remaining quarters of 1982, the bond yield declined to 14.51 percent, 13.75 percent and 11.88 percent.9 The pattern in the second half of 1982 is consistent with a decline in velocity. What remains unexplained, however, is the largest decline in velocity, which occurred in the first quarter."

If Tatom had had the right monetary aggregate he would have reached different conclusions. He would also have realised that if money velocity falls during most recessions it is because usually interest rates fall during recessions.

The mechanism relating a rise in interest rate to a rise in velocity (or as it has been sometimes called, the case of the missing money), is quite simple. But for economists who have been brought up to view money in a particular way it is very difficult to grasp.

Category: Economics


Philip George
Debunker of Keynesian, monetarist and Austrian economics

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