OccupySydneyZine 2012 Original Content Marc Ahrens

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Original

Contentby Marc Ahrens

ALEC, Big Pharma and theMedicalisation of Poverty

A Mathematical argument

for the existence of

climate change

Language, Jargon

Pantomimes and

Questions of Real Rigour

OCCUPYSYDNEYMarc Ahrens

Issue

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[email protected]

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ALEC, Big Pharma, and the Medicalisaon of Poverty

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A mathematical argument for the existence climate change.Marc Ahrens

Mathematics is special among sciences because indeed it is not truly a sciencebecause it is the only science in which assertions can actually be proven. As the

the most famous philosopher of science, Karl Popper1, made explicit, true sciences cannotassert things to be true, only not yet disproved, where disproving takes the form ofa physical prediction from the scientific theory which does not agree with the real world.

Ultimately the value of mathematics in sciences derives from the remarkable propensityfor nature to follow mathematics so consistently. Considering that the universe existedand its clockwork ticked long before any kind of maths to describe it, though, it becomesmuch less surprising that mathematics emerge to describe a particular set of rules followedby nature. All nature need do is follow them consistently, and someone will try to char-acterise this consistency. When a theory breaks down its not because statements withinthe particular mathematical framework have broken, but rather the wrong mathematical

model was chosen. The failure of Newtonian mechanics at speeds near lights was not asign that the mathematics of Newton was broken, but that the correspondence betweenthe physical world and Newtons mathematics was less ideal than the correspondence ofEinstein. The natural applicability of mathematics to natures rules, as long as it enactsthem consistently, means that science often becomes the process of figuring out what theunderlying mathematics are.

This sycophancy for mathematics is here to try and convince the unconvinced of thevalue of a mathematical argument supporting the existence of global warming. Whatfollows is not too hard to understand, so please be brave and bear with some maths.

The measurementsEach year the CSIRO and the Bureau of Meteorology publish a State of the Climatereport. State of the Climate 20122 says quite clearly, on page 10 beneath the headingAustralia in the context of global warming, that the

worlds 13 warmest years have all occurred in the 15 years since 1997.

How likely is this sequence of record-breaking hottest years? Meteorologists have takenphysical measurements, which we take at face value and try to connect to mathematics.

The mathematics of breaking recordsThe mathematics of record-breaking is well understood, and in fact it is linked to someof mathematicians most glorious work. In considering the likelihood of an event, wecan explore what an expected outcome is. Expected values can be arrived at pre-cisely, though the precision hangs on that of of the physical assumptions from which themathematicians start their work. Expected values are the most likely values of somephenomenon. The expected measured value of a phenomenon, and the changing likelihood

1Anon. Wikipedia page on Karl Popper. In: http://en.wikipedia.org/wiki/KarlPopper ().2CSIRO. State of the Climate. In: http://www.csiro.au/Outcomes/Climate/Understanding/State-

of-the-Climate-2012.aspx (2012).

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of particular measurements as one moves away from the expected ones is encapsulated bythe phenomenons statistical distribution.

What is the most likely expected number of record hottest years after one year ofrecord keeping? This we can say exactly is 1; as the only year on record, it is guaranteedto be the hottest. After one year there is always exactly one hottest year on record.

In the second year, the temperature can be either hotter or colder than the first, sothere is a 50% chance, a probability of 1

2, that the year is hotter than the first, a new

record. Thus, half the time there will be another hottest year to add to to the first, so theexpected number of hottest years is 1+ 1

2. In other words, if you do lots of record-keeping

experiments, or more simply, if you go through lots of different types of pre-existing datalooking for record-breaking measurements hottest years, wettest years, etc, etc andfrom every experiment you note how many record years thered been by the second year,and then you averaged these numbers, the figure youd get is one and a half.

The third year will produce a temperature, c. Calling the previous two temperaturesa & b (year 1 and 2 resp.), then consider all the possible orderings, where temperatures

are order by size: a < b < c, b < a < c, a < c < b, b < c < a, c < a < b,and c < b < a.

There are six possible orderings, two of which have c as the highest temperature. Thus,there is a 2

6, ie 1

3, chance that the third year is hotter than the other two, making it a

new record. Thus, after 3 years, the expected number of hottest years is 1 + 12

+ 13

= 116

one and five sixths.Extrapolating, after n years, the expected number of record years is 1+ 1

2+ 1

3+ . . . + 1

n.

Taking this series to be infinitely long (ie n ) gives the famous Harmonic Series,named as such because each term in the series corresponds to another overtone on thestring of a musical instrument, say, or a pipe in an organ. The partial sums are denoted

for convenience as Hn, with H1 = 1, H2 = 1 +1

2 , H3 = 1 +1

2 +1

3 , etc. These sums addto numbers that are known as the Harmonic Numbers, and notice: the nth Harmonicnumber Hn is the expected number of hottest years after n years.

Asides: First note that the full harmonic series is not actually a number; it does notconverge, as mathematicians say. Instead, adding more and more terms just makes thesum bigger and bigger, so that with infinitely many terms, the sum is infinitely large.This might sound obvious, but there are infinite series whose sum is less than infinity, iethey sum to numbers, such as any sum r0 + r1 + r2 + r3 + ... where r < 1, which equals1

1r. Note also that the harmonic series is a special case of the Riemman Zeta Function:

its

(r) 1 + 12r

+ 13r

+ 14r

+ . . . (1)

with r = 1. This function is famous among mathematicians because of its role in link-ing complex analysis, which is a system of mathematics so general it can understandsquare-roots of negative numbers, to number theory, the mathematics of the naturalnumbers, {0, 1, 2, 3, 4, . . .}. The connection afforded by the Zeta function allows proofsin number theory, such as that of Fermats (famous) Last Theorem, to be constructed interms of results from complex analysis.

For more details of the Harmonic Series, Riemmans Zeta Function, the finer pointsof logarithms and exponential functions, a historiography of post-Enlightenment mathe-

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matics, and so much more, see the excellent recreational maths book by Havil3. It wasthere that I was first alerted to the the record-breaking argument for climate change.

Values of Hn

Using a tiny bit of program code, one can generate the Harmonic Numbers, all but thefirst of which are fractions. The decimal approximations are also given below:

nHn, expected number ofrecord hottest years after nyears of record keeping

1 12 3/2 (1.5)3 11/6 (approx. 1.833333)4 25/12 (approx. 2.083333)5 137/60 (approx. 2.283333)6 49/20 (2.45)7 363/140 (approx. 2.592857)8 761/280 (approx. 2.717857)9 7129/2520 (approx. 2.828968)

10 7381/2520 (approx. 2.928968)11 83711/27720 (approx. 3.019877)12 86021/27720 (approx. 3.103211)13 1145993/360360 (approx. 3.180134)

A related sequence of numbers can be obtained, call it In

, where In

is the number of

years of record keeping required until the expected number of records exceeds n. Lookingat the above values of H

n, we can see that I1 = 1, I2 = 4 & I3 = 11. Furthermore:

n

In

, expected number ofyears of record-keeping inorder to observe your nthrecord-breaking year

1 12 43 114 31

5 836 2277 6168 16749 4550

10 1236711 3361712 9138013 248397

3

J Havil. Gamma: Exploring Eulers Constant. Princeton, NJ: Princeton University Press, 2003.

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These figures can be produced with a small program, or by consulting a resource suchas N. J. A. Sloans The On-Line Encyclopedia of Integer Sequences4, where this sequenceis indexed as A004080.

Note that the last line says that the expected number of years it should take to see 13record-breaking years is 248,397.

Hottest years an anomaly

With the above in mind, we can probe the likelihood of having 13 of the last 15 years ashottest years on record. By this stage, you mightve already spotted a disagreementbetween the mathematics of record-breaking and the CSIRO claims. It is this: the CSIROclaims that 13 of the last 15 years have each been record-breakers, whereas the sequenceIn

(Sloans A0040805) tells us that the expected number of years to wait to see 13 record-breaking years is 248397. It should take about 250 thousand years to experience 13record-hottest years, yet it has taken only 15.

How does the discrepancy occur? While introducing the mathematics of record break-ing and the Harmonic Numbers, one vital fact was omitted: this mathematics only applieswhen the underlying process is not changing its nature, so that its statistical qualities arenot changing over time. More technically, these mathematics of record breaking have beenproven to work only for statistical processes whose distribution remain fixed6. In suchcases, as the sequence I

nsuggests, records should become harder and harder to break,

as each new record becomes more unattainable for challengers unless, of course, theunderlying process is changing in such a way that new records are more likely.

In the case of the 100m sprint records, if nothing changed about the human bodyand the nature of the competition then new records would be very rare by now. Instead,

sports science, the professionalisation of sport endowing it with more money to pursuenew advantages, better shoes, more aerodynamic clothes, and perhaps more sophisticateddrug cheating continue to enable new records.

Conclusion

In our case, it appears the correspondence between the real world an our mathematicalmodel has broken down. But! This is actually useful in this case, for as Ned Glick hassuggested7, we can infer something from this breakdown about the source of our data:that, unlike the mathematical model expects, its statistics are indeed changing over time.In other words, the fact that the mathematics of record years suggests 13 record-breaking

years occur in about 250,000, whereas the climate has taken only 15 years to produce 13record-breakers, is extremely unlikely to occur for any reason other than an upward trendin the underlying statistics. Assuming the initial measurements are correct, its very likelythe climate is warming.

4N.J.A. Sloan. The On-Line Encyclopedia of Integer Sequences. In: http://oeis.org/A004080/list ().5Ibid.6For the more mathematically inclined, the measurements must come from an independent and iden-

tically distributed (IID) process. The remarkable fact is that the Harmonic Numbers give the expected

number of record years, independently of the underlying distribution.7N. Glick. Breaking records and breaking boards. In: Am. Math. Mon. 85, 2, 26. (1978).

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Appendix: Keens problem with the mathematics of

competition1

Consider a firm producing in a marketplace, the ith among many. Neoclassical economics

requires for its claims here Perfect Competition, namely that (1) there are many firmsso that none of have a large market share, lest their production quantity influence themarket price, and (2) firms produce identical products (commodities), so consumershave no reason not to follow the cheapest price2.

Under such conditions, there will be a market-wide price, P.The ith firm will have a profit i which depends on the quantity produced and sold

by the firm (qi), the market price, and the cost of manufacturing qi items, Ci, which isparticular to the processes of firm i.

By definition, profit for firm i is the revenue P qi minus the cost of production Ci:

i(qi) Pqi Ci

As per high school maths, the quantity qi that maximizes profit i(qi) is the one thatmakes its derivative with respect to qi zero.

0 = i(qi) =d(Pqi)dqi

Ci(qi) (1)

Now, at this point neoclassical economists use the Perfect Competition conditionsto assert that P is independent of qi the ith firm is too small to influence the marketwith its quantity qi, so that

d(Pqi)dqi

= Pdqidqi

= P. This gives them the result that profitsare maximised when

0 = i(q

i)

?= P C

i(q

i) (Standard result contestded by Keen) (2)

In other words, Profits are maximum when the Ci(qi), the marginal cost, equalsthe price. This is the optimum output rule for a firm in perfect competition 3.

Instead of settling there, lets continue expanding the mathematics from equation (1).

0 = Pdqidqi

+ qidPdqi Ci(qi) (Product rule)

= P+ qidPdqi Ci(qi)

Compare this with equation (2) and note that the two are the same exactly whenqi

dPdqi

= 0. Again, neoclassical economists evoke perfect competition to assert this astrue in order to get their standard result, since P is independent of qi if no firm has

an influential market share, meaningdPdqi , and hence the whole term, is zero. But lets

continue expanding the maths...

0 = P+ qidPdQ

dQ

dqi Ci(qi) (Chain rule)

1S Keen and R Standish. Debunking the theory of the firm. In: Real World Economics Review, Iss.53 (2010). url: http://www.paecon.net/PAEReview/issue53/KeenStandish53.pdf.

2Paul Krugman, Robin Wells, and Kathryn Graddy. Essentials of Economics. 2nd. Worth Publishers,NY, 2010.

3Ibid.

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where Q is the total output of the commodity in question of all firms, ie Q =

j qj. Thus,

0 = P+ qidPdQ

d(

j qj)dqi

Ci(qi) (Def of Q)

= P+ qidPdQ

j

dqjdqi

Ci(qi)

= P+ qi dPdQ

j i,j Ci(qi) (Def of Kronekcer Delta)

= P+ qidPdQ Ci(qi) (i,j = 1 when i = j and 0 otherwise)

The last equation above is mathematically equivalent to the equation whose zeros give themaximum of the very first one (ie to eq. (1)). Comparing again with eq. (2) suggests that,if the two are to be equivalent if the standard result stated for a competitive firm is tobe derivable from the above equation, which is equivalent to eq. (1) then it requiresthat qi

dPdQ

= 0. Now, it cannot be that qi is naught: manufacturers must manufacturer

something. Thus, the only way the standard result can be true is if dPdQ

= 0.

And here we arrive at a contradiction, for dPdQ

is the slope of the demand curve for the

whole market place, and another central tenet of neoclassical economics is that this mustalways be negative, that is, always downward sloping. It cannot be zero. For neoclassicists,dPdqi

can certainly be zero indeed, they insist it is, as a result of perfect competition, but

not dPdQ

. What the derivation above has demonstrated is that these two expressions aremathematically equivalent. It is a failure of the mathematics of neoclassical economistsinsight they have different values.

In different arguments elsewhere4, Keen relays other reasons for the failure of theOptimum Output Rule, which was first demonstrated by William Gorman in 19535

who completely failed to recognise what he had proven, and was then rediscovered andpublicised in the 70s as the Sonnenschein-Mantel-Debreu Theorem67 . SMD says that the

demand curve for a whole marketplace cannot be downward sloping at all points exceptwhen there is only one consumer and one commodity; the market-wide demand curve cantmathematically exist as traditionally conceived. But even the worrisome SMD result iscompatible with a market-wide demand curve that is mostly downward sloping; the abovederivation suggests that, for neoclassical economists Optimum Output Rule in perfectcompetition, the market-wide demand curve must be never downward sloping, nor indeedever upward sloping.

The failure of the Optimum Output Rule presents a moral problem for economics,for whom price is related to utility the benefit derived from a good. Jeremy Benthamfounded economics with his work on utilitarianism, the ethical principle that the most

ethical course is one that maximises benefit (utility) across society The greatestgood for the greatest number. This hinges on a measure of utility. Bentham decidedthat a suitable measure of the utility of a product is what people are willing to pay forit; when people spend money on an item, they affirm their belief that the item is worthat least that much. Thus, the promise of the Optimum Output Rule is that it doesntmerely maximise profits, but that it maximises the benefit to society. With that rulegone, so is the moral justification for the basis of laissez faire capitalism.

4Steve Keen. Debunking Economics, 2nd. London: Zen Books, 2010.5W M Gorman. Community preference fields. In: Econometrica (1953).6Wikipedia. The Sonnenschein-Mantel-Debreu Theorem. url: http://en.wikipedia.org/wiki/

Sonnenschein%E2%80%93Mantel%E2%80%93Debreu_theorem.7

Frank Ackerman. Still dead after all these years: interpreting the failure of general equilibriumtheory. In: Journal of Economic Methodology 9:2, 119139 (2002). url: http://www.ase.tufts.edu/gdae/Pubs/rp/StillDead02.pdf.

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