Tag Archives: mathematics

La historia de π (pi)

π is wrong!” by Bob Palais

appeared in

The Mathematical Intelligencer Springer-Verlag New York Volume 23, Number 3, 2001, pp. 7-8.

Bob Palais gratefully acknowledges Dr. Chandler Davis, for his encouragement and editorial input.

(See also the Wikipedia entry on Dr. Davis.) The most amusing letter to the editor in response stated:

“I agree with Bob Palais’ pi-ous article, but it may be 2-pi-ous.”

The Pi Manifesto


Last updated July 4th, 2011

– See more at: http://www.thepimanifesto.com/#sthash.R4ubVZpw.dpuf

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Science, Technology, Engineering and Mathematics (STEM, previously SMET) is an acronym that refers to the academic disciplines of science[note 1], technology, engineeringand mathematics.[1] The term is typically used when addressing education policy and curriculum choices in schools to improve competitiveness in science and technology development. It has implications for workforce development, national security concerns and immigration policy.[1] Education emphasizing STEM disciplines is considered to be more beneficial to the student than the previous generation of education standards that emphasizes broad “core” disciplines and social skills instead.

The acronym arose in common use shortly after an interagency meeting on science education held at the US National Science Foundation chaired by the then NSF director Rita Colwell.[citation needed] A director from the Office of Science division of Workforce Development for Teachers and Scientists, Dr. Peter Faletra, suggested the change from the older acronym SMET to STEM. Dr. Colwell, expressing some dislike for the older acronym, responded by suggesting NSF to institute the change. One of the first NSF projects to use the acronym was STEMTEC, the Science, Technology, Engineering and Math Teacher Education Collaborative at the University of Massachusetts Amherst, which was funded in 1997.

axioms and dogmas

Axioms are self evident truths that require no proof, which is similar to a dogmatic belief in the sense that dogma is a set of beliefs or doctrines that are established as undoubtedly in truth.

Axiom is a statement taken to hold within a particular theory. One can combine the axioms to prove things within that theory. One may add or remove axioms to the theory to get another theory.

An axiom is something that is self-evidently true; it is so obvious that there is no controversy about it. In mathematics, you just have to accept some very basic notions in order to avoid circular reasoning. These can’t be proven, but they can always (and often very easily) be observed.

Example from Euclid’s Elements:

Common notions:

Things that are equal to the same thing are also equal to one another (Transitive property of equality).
If equals are added to equals, then the wholes are equal.
If equals are subtracted from equals, then the remainders are equal.
Things that coincide with one another equal one another (Reflexive Property).
The whole is greater than the part.

Dogmas are axioms of cultural, religious, political belief systems.

A dogma refers to (usually a religious) teaching that is considered undoubtedly and absolutely true. It is something you accept without any direct observation; dogmas are accepted by faith only.

Some people would say that there is no difference between axioms and dogmas, because ‘self-evident truths’ are in some sense based on faith; that is that you accept on faith that anything that seems obvious and self-evident is true. An interesting read on this subject is Wittgenstein’s On Certainty. Take the axiom of choice, for example: there is huge division in mathematics as to whether it’s true or not, and many proofs are written based on a by-faith acceptance (or rejection) of said axiom.

The difference is that it is perfectly ok to handle different sets of axioms in, say, mathematics and prove a theorem in Euclidean geometry one day and a theorem in Lobachevskian the next – just remembering when the fifth postulate does or doesn’t hold, but it’s not considered acceptable to hold several sets of dogmas at once.

Simpson’s paradox

In probability and statistics, Simpson’s paradox, or the Yule–Simpson effect, is a paradox in which a trend that appears in different groups of data disappears when these groups are combined, and the reverse trend appears for the aggregate data. This result is often encountered in social-science and medical-science statistics,[1] and is particularly confounding when frequency data are unduly given causal interpretations.[2] Simpson’s Paradox disappears when causal relations are brought into consideration. Many statisticians believe that the mainstream public should be informed of the counter-intuitive results in statistics such as Simpson’s paradox.[3][4]

Edward H. Simpson first described this phenomenon in a technical paper in 1951,[5] but the statisticians Karl Pearson, et al., in 1899,[6] and Udny Yule, in 1903, had mentioned similar effects earlier.[7] The name Simpson’s paradox was introduced by Colin R. Blyth in 1972.[8] Since Edward Simpson did not actually discover this statistical paradox (an instance of Stigler’s law of eponymy), some writers, instead, have used the impersonal names reversal paradox and amalgamation paradox in referring to what is now called Simpson’s Paradox and the Yule–Simpson effect.


Kidney stone treatment

This is a real-life example from a medical study[10] comparing the success rates of two treatments for kidney stones.[11]

The table below shows the success rates and numbers of treatments for treatments involving both small and large kidney stones, where Treatment A includes all open surgical procedures and Treatment B is percutaneous nephrolithotomy. The numbers in parentheses indicate the number of success cases over the total size of the group. (For example, 93% equals 81 divided by 87.)

Treatment A Treatment B
Small Stones Group 1
93% (81/87)
Group 2
87% (234/270)
Large Stones Group 3
73% (192/263)
Group 4
69% (55/80)
Both 78% (273/350) 83% (289/350)

The paradoxical conclusion is that treatment A is more effective when used on small stones, and also when used on large stones, yet treatment B is more effective when considering both sizes at the same time. In this example the “lurking” variable (or confounding variable) of the stone size was not previously known to be important until its effects were included.

Which treatment is considered better is determined by an inequality between two ratios (successes/total). The reversal of the inequality between the ratios, which creates Simpson’s paradox, happens because two effects occur together:

  1. The sizes of the groups, which are combined when the lurking variable is ignored, are very different. Doctors tend to give the severe cases (large stones) the better treatment (A), and the milder cases (small stones) the inferior treatment (B). Therefore, the totals are dominated by groups 3 and 2, and not by the two much smaller groups 1 and 4.
  2. The lurking variable has a large effect on the ratios, i.e. the success rate is more strongly influenced by the severity of the case than by the choice of treatment. Therefore, the group of patients with large stones using treatment A (group 3) does worse than the group with small stones, even if the latter used the inferior treatment B (group 2).

Based on these effects, the paradoxical result can be rephrased more intuitively as follows: Treatment A, when applied to a patient population consisting mainly of patients with large stones, is less successful than Treatment B applied to a patient population consisting mainly of patients with small stones.

Berkeley gender bias case

One of the best-known real-life examples of Simpson’s paradox occurred when the University of California, Berkeley was sued for bias against women who had applied for admission to graduate schools there. The admission figures for the fall of 1973 showed that men applying were more likely than women to be admitted, and the difference was so large that it was unlikely to be due to chance.[12][13]

Applicants Admitted
Men 8442 44%
Women 4321 35%

But when examining the individual departments, it appeared that no department was significantly biased against women. In fact, most departments had a “small but statistically significant bias in favor of women.”[13] The data from the six largest departments are listed below.

Department Men Women
Applicants Admitted Applicants Admitted
A 825 62% 108 82%
B 560 63% 25 68%
C 325 37% 593 34%
D 417 33% 375 35%
E 191 28% 393 24%
F 373 6% 341 7%

The research paper by Bickel et al.[13] concluded that women tended to apply to competitive departments with low rates of admission even among qualified applicants (such as in the English Department), whereas men tended to apply to less-competitive departments with high rates of admission among the qualified applicants (such as in engineering and chemistry). The conditions under which the admissions’ frequency data from specific departments constitute a proper defense against charges of discrimination are formulated in the book Causality by Pearl.[2]

Low birth weight paradox

The low birth weight paradox is an apparently paradoxical observation relating to the birth weights and mortality of children born to tobacco smoking mothers. As a usual practice, babies weighing less than a certain amount (which varies between different countries) have been classified as having low birth weight. In a given population, babies with low birth weights have had a significantly higher infant mortality rate than others. normal birth weight infants of smokers have about the same mortality rate as normal birth weight infants of non-smokers, and low birth weight infants of smokers have a much lower mortality rate than low birth weight infants of non-smokers, but infants of smokers overall have a much higher mortality rate than infants of non-smokers. This is because many more infants of smokers are low birth weight, and low birth weight babies have a much higher mortality rate than normal birth weight babies.[14]

Batting averages

A common example of Simpson’s Paradox involves the batting averages of players in professional baseball. It is possible for one player to hit for a higher batting average than another player during a given year, and to do so again during the next year, but to have a lower batting average when the two years are combined. This phenomenon can occur when there are large differences in the number of at-bats between the years. (The same situation applies to calculating batting averages for the first half of the baseball season, and during the second half, and then combining all of the data for the season’s batting average.)

A real-life example is provided by Ken Ross[15] and involves the batting average of two baseball players, Derek Jeter and David Justice, during the baseball years 1995 and 1996:[16]

1995 1996 Combined
Derek Jeter 12/48 .250 183/582 .314 195/630 .310
David Justice 104/411 .253 45/140 .321 149/551 .270

In both 1995 and 1996, Justice had a higher batting average (in bold type) than Jeter did. However, when the two baseball seasons are combined, Jeter shows a higher batting average than Justice. According to Ross, this phenomenon would be observed about once per year among the possible pairs of interesting baseball players. In this particular case, the Simpson’s Paradox can still be observed if the year 1997 is also taken into account:

1995 1996 1997 Combined
Derek Jeter 12/48 .250 183/582 .314 190/654 .291 385/1284 .300
David Justice 104/411 .253 45/140 .321 163/495 .329 312/1046 .298

The Jeter and Justice example of Simpson’s paradox was referred to in the “Conspiracy Theory” episode of the television series Numb3rs, though a chart shown omitted some of the data, and listed the 1996 averages as 1995.[citation needed]

If you use weighting this goes away. Normalise for the largest totals so that you are comparing the same thing.

1995 1996 Combined
Derek Jeter 12/48*411 102.75/411 .250 183/582*582 183/582 .314 285.75/993 .288
David Justice 104/411*411 104/411 .253 45/140*582 187/582 .321 291/993 .293

David Hilbert

David Hilbert (German: [ˈdaːvɪt ˈhɪlbɐt]; January 23, 1862 – February 14, 1943) was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry. He also formulated the theory of Hilbert spaces,[3] one of the foundations of functional analysis.

Hilbert adopted and warmly defended Georg Cantor‘s set theory and transfinite numbers. A famous example of his leadership in mathematics is his 1900 presentation of a collection of problems that set the course for much of the mathematical research of the 20th century.

Hilbert and his students contributed significantly to establishing rigor and developed important tools used in modern mathematical physics. Hilbert is known as one of the founders of proof theory and mathematical logic, as well as for being among the first to distinguish between mathematics and metamathematics.

Hilbert lived to see the Nazis purge many of the prominent faculty members at University of Göttingen in 1933.[14] Those forced out included Hermann Weyl (who had taken Hilbert’s chair when he retired in 1930), Emmy Noether and Edmund Landau. One who had to leave Germany, Paul Bernays, had collaborated with Hilbert in mathematical logic, and co-authored with him the important book Grundlagen der Mathematik (which eventually appeared in two volumes, in 1934 and 1939). This was a sequel to the Hilbert-Ackermann book Principles of Mathematical Logic from 1928.

About a year later, Hilbert attended a banquet and was seated next to the new Minister of Education, Bernhard Rust. Rust asked, “How is mathematics in Göttingen now that it has been freed of the Jewish influence?” Hilbert replied, “Mathematics in Göttingen? There is really none any more.”[15]

Hilbert’s tomb:
Wir müssen wissen
Wir werden wissen

By the time Hilbert died in 1943, the Nazis had nearly completely restaffed the university, inasmuch as many of the former faculty had either been Jewish or married to Jews. Hilbert’s funeral was attended by fewer than a dozen people, only two of whom were fellow academics, among them Arnold Sommerfeld, a theoretical physicist and also a native of Königsberg.[16] News of his death only became known to the wider world six months after he had died.

Hilbert was baptized and raised in the Reformed Protestant Church.[17] He later on left the Church and became an agnostic.[18] He also argued that mathematical truth was independent of the existence of God or other a priori assumptions.[19][20]

The epitaph on his tombstone in Göttingen consists of the famous lines he spoke at the conclusion of his retirement address to the Society of German Scientists and Physicians in the autumn of 1930. The words were given in response to the Latin maxim: “Ignoramus et ignorabimus” or “We do not know, we shall not know”:[21]

Wir müssen wissen.
Wir werden wissen.

In English:

We must know.
We will know.

The day before Hilbert pronounced these phrases at the 1930 annual meeting of the Society of German Scientists and Physicians, Kurt Gödel—in a roundtable discussion during the Conference on Epistemology held jointly with the Society meetings—tentatively announced the first expression of his incompleteness theorem.[22]