Tag Archives: Mathematics

Power

I have rewritten the entry to “power” in the abstractmath.org Glossary:

POWER

Here are three variant phrases that say that $125=5^3$:

  • “$125$ is a power of $5$ with exponent $3$”.
  • “$125$ is the third power of $5$”.
  • “$125$ is $5$ to the third power”.

Some students are confused by such statements, and conclude that $3$ is the “power”. This usage appears in print in Wikipedia in its entry on Exponentiation (as it was on 22 November 2016):

“…$b^n$ is the product of multiplying $n$ bases:

\[b^n = \underbrace{b \times \cdots \times b}_n\]

In that case, $b^n$ is called the $n$-th power of $b$, or $b$ raised to the power $n$.”

As a result, students (and many mathematicians) refer to $n$ as the “power” in any expression of the form “$a^n$”. The number $n$ should be called the “exponent”. The word “power” should refer only to the result $a^n$. I know mathematical terminology is pretty chaotic, but it is silly to refer both to $n$ and to $a^n$ as the “power”.

Almost as silly as using $(a,b)$ to refer to an open interval, an ordered pair and the GCD. (See The notation $(a,b)$.)

Suggestion for lexicographical research: How widespread does referring to $n$ as the “power” come up in math textbooks or papers? (See usage.)

Thanks to Tomaz Cedilnik for comments on the first version of this entry.

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The great math mystery

The great math mystery

Last night Nova aired The great math mystery, a documentary that describes mathematicians’ ideas about whether math is discovered or invented, whether it is “out there” or “in our head”. It was well-done. Things were explained clearly using images and metaphors, although they did show Maxwell’s equations as algebra (without explaining it). The visual illustrations of connections between Maxwell’s equations and music and electromagnetic waves was one of the best parts of the documentary.

In my opinion they made good choices of mathematical ideas to cover, but I imagine a lot of research mathematicians will have a hissy that they didn’t cover XXX (their subject).

The applications to physics dominated the show (that is not a complaint), but someone did mention the remarkable depth of number theory. Number theory is deep pure math that has indeed had some applications, but that’s not why some of the greatest mathematicians in the world have spent their lives on the subject. I believe logic and proof was never mentioned, and that is completely appropriate for a video made for the general public. Some mathematicians will disagree with that last sentence.

Where does math live?

The question,

Does math live

  • In an ideal world separate from the physical world,
  • in the physical world, or
  • in our brains?

has a perfectly clear answer: It exists in our brains.

Ideal world

The notion that math lives in an ideal world, as Plato supposedly believed, has no evidence for it at all.

I suppose you could say that Plato’s ideal world does exist — in our brains. But that wouldn’t be quite correct: We have a mental image of Plato’s ideal world in our brains, but that image is not the whole ideal world: If we know about triangles, we can imagine the Ideal Triangle to be in his world, but we have to know about the zeta function or the monster group to visualize them to be in his world. Even then, the monster group in our brain is just a collection of neurons connected to concepts such as “largest sporadic simple group” or “contains\[2^{46} \cdot 3^{20} \cdot 5^9 \cdot 7^6 \cdot 11^2 \cdot 13^3 \cdot 17 \cdot 19 \cdot 23 \cdot 29 \cdot 31 \cdot 41 \cdot 47 \cdot 59 \cdot 71\]elements” — but there is not a neuron for each element! We don’t have that many neurons.

The size of the monster group does not live in my brain. I copied it from Wikipedia.

Real world

Our collective experience is that math is extraordinarily useful for modeling many aspects of the real world. But in what sense does that mean it exists in the real world?

There is a sense in which a model of the real world exists in our brains. If we know some of the math that explains certain aspects of the real world, our brains have neuron connections that make that math live in our brain and in some sense in the model of the real world that is in our brain. But does that mean the math is “out there”? I don’t see why.

Math is a social endeavor

One point that usually gets left out of discussions of Platonism is this: Some math exists in any individual person’s brain. But math also exists in society. The math floating around in the individual brains of people is subject to frequent amendments to those people’s understanding because they interact with the real world and in particular with other people.

In particular, theoretical math exists in the society of mathematicians. It is constantly fluctuating because mathematicians talk to each other. They also explain it to non-mathematicians, which as everyone know can bring new insights into the brain of the person doing the explaining.

So I think that the best answer to the question, where does math live? is that math is a bunch of memes that live in our social brain.

References

I have written about these issues before:

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Pattern recognition and me

Recently, I revised the abstractmath.org article on pattern recognition. Doing that that prompted me to write about my own experiences with patterns. Recognizing patterns is something that has always delighted me: it is more of a big deal for me than it does for many other people. That, I believe, is what led me into doing research in math.

I have had several experiences with déjà vu, which is the result of pattern recognition with one pattern hidden. That will be a separate post. I expect to post about my experiences in recognizing patterns in math as well.

Patterns in language

As a teenager I was a page in the Savannah Public Library. There I discovered grammars for many languages. The grammars of other languages are astonishingly different from each other and are full of obscurities that I love to detect. Until I went to college, I was the only person I knew who read grammars for fun.

I am using the word “grammar” in the sense that linguists use it: patterns in our speech and writing, mostly unnoticed, that help express what we want to say)

The word “grammar” is also used to mean rules laid down by the ruling classes about phrases like “between you and I” and the uses of “whom”. Such rules primarily divide the underprivileged from the privileged, and many will disappear when the older members of the privileged class die (but they will think of new ones).

Grammar-induced glee

Russian

I got pretty good at reading and speaking Russian when I was a student (1959-62), but most of it has disappeared. In 1990, we hosted a Russian cello student with the Soviet-American Youth Orchestra for a couple of days. I could hardly say anything to him. One time he noticed one of our cats and said “кошка”, to which I replied “два кошки” (“two cats”). He responded by correcting me: “две кошки”. Then I remembered that the word for “two” in Russian is the only word in the language that distinguishes gender in the plural. I excitedly went around telling people about this until I realized that no one cared.

Spanish

Recently I visited a display about the Maya at the Minnesota Science Museum that had all its posters in English and Spanish. I discovered a past subjunctive in one of the Spanish texts. That was exciting, but I had no one to be excited with.

The preceding paragraph is an example of a Pity Play.

Just the other day our choir learned a piece for Christmas with Spanish words. It had three lines in a row ending in a past subjunctive. (It is in rhyming triples and if you use all first conjugation verbs they rhyme.) Such excitement.

Turkish

During the Cold War, I spent 18 months at İncirlik Air Base in Turkey. Turkish is a wonderful language for us geeks, very complicated yet most everything is regular. Like a computer language.

I didn’t know about computer languages during the Cold War, although they were just beginning to be used. I did work on a “computer” that you programmed by plugging cables into holes in various ways.

In Turkish, to modify a noun by a noun, you add an ending to the second noun. “İş Bankası” (no dot over the i) means “business bank”. (We would say “commercial bank”.) “İş” means “business” and “bank” by itself is “banka”. Do you think this is a lovably odd pattern? Well I do. But that’s the way I am.

A spate of spit

We live a couple blocks from Minnehaha Falls in Minneapolis. Last June the river flooded quite furiously and I went down to photograph it. I thought to my self, the river is in full spate. I wondered if the word “spate” came from the same IE root as the word “spit”. I got all excited and went home and looked it up. (No conclusion –it looks like it might be but there is no citation that proves it). Do you know anyone who gets excited about etymology?

Secret patterns in nature

All around us there are natural patterns that people don’t know about.

Cedars in Kentucky

For many years, we occasionally drove back and forth between Cleveland (where we lived) and Atlanta (where I had many relatives). We often stopped in Kentucky, where Jane grew up. It delighted me to drive by abandoned fields in Kentucky where cedars were colonizing. (They are “red cedars,” which are really junipers, but the name “cedar” is universal in the American midwest.)

What delighted me was that I knew a secret pattern: The presence of cedars means that the soil is over limestone. There is a large region including much of Kentucky and southern Indiana that lies over limestone underneath.

That gives me another secret: When you look closely at limestone blocks in a building in Bloomington, Indiana, you can see fossils. (It is better if the block is not polished, which unfortunately the University of Indiana buildings mostly are.) Not many people care about things like this.

The bump on Georgia

The first piece of pattern recognition that I remember was noticing that some states had “bumps”. This resulted in a confusing conversation with my mother. See Why Georgia has a bump.

Maybe soon I will write about why some states have panhandles, including the New England state that has a tiny panhandle that almost no one knows about.

Minnesota river

We live in Minneapolis now and occasionally drive over the Mendota Bridge, which crosses the Minnesota River. That river is medium sized, although it is a river, unlike Minnehaha Creek. But the Minnesota River Valley is a huge wide valley completely out of proportion with its river. This peculiarity hides a Secret Story that even many Minnesotans don’t know about.

The Minnesota River starts in western Minnesota and flows south and east until it runs into the Mississippi River. The source of the Red River is a few miles north of the source of the Minnesota. It flows north, becoming the boundary with North Dakota and going by Fargo and through Winnipeg and then flows into Lake Winnipeg. Thousands of years ago, all of the Red River was part of the Minnesota River and flowed south, bringing huge amounts of meltwater from the glaciers. That is what made the big valley. Eventually the glaciers receded far enough that the northern part of the river changed direction and started flowing north, leaving the Minnesota River a respectable medium sized river in a giant valley.

The Mendota Bridge is also one of the few places in the area where you can see the skyscrapers of Minneapolis and of St Paul simultaneously.

Music

Baroque music

I love baroque music because of patterns such as fugues, which I understood, and the harmony it uses, which I still don’t understand. When I was 10 years old I had already detected its different harmony and asked my music teacher about it. She waved her hands and declaimed, “I don’t understand Bach.” (She was given to proclamations. Once she said, “I am never going out of the State of Georgia again because in Virginia they put mayonnaise on their hamburgers!”)

Some baroque music uses a ground bass, which floored me when I first heard it. I went on a rampage looking for records of chaconnes and passacaglias. Then I discovered early rock music (Beatles, Doors) and figured out that they sometimes used a ground bass too. That is one of the major attractions of rock music for me, along with its patterns of harmony.

Shape note music

Some shape note tunes (for example, Villulia), as well as some early rock music, has a funny hollow sound that sounds Asian to me. I delight in secretly knowing why: They use parallel fifths.

The Beatles have one song (I have forgotten which) that had a tune which in one place had three or four beats in a row that were sung on the same pitch — except once, when the (third I think) beat was raised a fourth. I fell in love with that and excitedly pointed it out to people. They looked at me funny. Later on, I found several shape note tunes that have that same pattern.

Links

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The Greek alphabet in math

This is a revision of the portion of the article Alphabets in abstractmath.org that describes the use of the Greek alphabet by mathematicians.

Every letter of the Greek alphabet except omicron is used in math. All the other lowercase forms and all those uppercase forms that are not identical with the Latin alphabet are used.

  • Many Greek letters are used as proper names of mathe­ma­tical objects, for example $\pi$. Here, I provide some usages that might be known to undergraduate math majors.  Many other usages are given in MathWorld and in Wikipedia. In both those sources, each letter has an individual entry.
  • But any mathematician will feel free to use any Greek letter with a meaning different from common usage. This includes $\pi$, which for example is often used to denote a projection.
  • Greek letters are widely used in other sciences, but I have not attempted to cover those uses here.

The letters

  • English-speaking mathematicians pronounce these letters in various ways.  There is a substantial difference between the way American mathe­maticians pronounce them and the way they are pronounced by English-speaking mathe­maticians whose background is from British Commonwealth countries. (This is indicated below by (Br).)
  • Mathematicians speaking languages other than English may pronounce these letters differently. In particular, in modern Greek, most Greek letters are pro­nounced differ­ently from the way we pronounce them; β for example is pro­nounced vēta (last vowel as in "father").
  • Newcomers to abstract math often don’t know the names of some of the letters, or mispronounce them if they do.  I have heard young mathe­maticians pronounce $\phi $ and $\psi $ in exactly the same way, and since they were writing it on the board I doubt that anyone except language geeks like me noticed that they were doing it.  Another one pronounced $\phi $ as  “fee” and $\psi $ as “fie”.

Pronunciation key

  • ăt, āte, ɘgo (ago), bĕt, ēve, pĭt, rīde, cŏt, gō, ŭp, mūte.
  • Stress is indicated by an apostrophe after the stressed syllable, for example ū'nit, ɘgō'.
  • The pronunciations given below are what mathematicians usually use. In some cases this includes pronunciations not found in dictionaries.

 

Alpha: $\text{A},\, \alpha$: ă'lfɘ. Used occasionally as a variable, for example for angles or ordinals. Should be kept distinct from the proportionality sign "∝".

 

Beta: $\text{B},\, \beta $: bā'tɘ or (Br) bē'tɘ. The Euler Beta function is a function of two variables denoted by $B$. (The capital beta looks just like a "B" but they call it “beta” anyway.)  The Dirichlet beta function is a function of one variable denoted by $\beta$.

 

Gamma: $\Gamma, \,\gamma$: gă'mɘ. Used for the names of variables and functions. One familiar one is the $\Gamma$ function. Don’t refer to lower case "$\gamma$" as “r”, or snooty cognoscenti may ridicule you.


Delta: $\Delta \text{,}\,\,\delta$: dĕltɘ. The Dirac delta function and the Kronecker delta are denoted by $\delta $.  $\Delta x$ denotes the change or increment in x and $\Delta f$ denotes the Laplacian of a multivariable function. Lowercase $\delta$, along with $\epsilon$, is used as standard notation in the $\epsilon\text{-}\delta$ definition of limit.


Epsilon: $\text{E},\,\epsilon$ or $\varepsilon$: ĕp'sĭlɘn, ĕp'sĭlŏn, sometimes ĕpsī'lɘn. I am not aware of anyone using both lowercase forms $\epsilon$ and $\varepsilon$ to mean different things. The letter $\epsilon $ is frequently used informally to denoted a positive real number that is thought of as being small. The symbol ∈ for elementhood is not an epsilon, but many mathematicians use an epsilon for it anyway.


Zeta: $\text{Z},\zeta$: zā'tɘ or (Br) zē'tɘ. There are many functions called “zeta functions” and they are mostly related to each other. The Riemann hypothesis concerns the Riemann $\zeta $-function.


Eta: $\text{H},\,\eta$: ā'tɘ or (Br) ē'tɘ. Don't pronounce $\eta$ as "N" or you will reveal your newbieness.


Theta: $\Theta ,\,\theta$ or $\vartheta$: thā'tɘ or (Br) thē'tɘ.  The letter $\theta $ is commonly used to denote an angle. There is also a Jacobi $\theta $-function related to the Riemann $\zeta $-function. See also Wikipedia.


Iota: $\text{I},\,\iota$: īō'tɘ. Occurs occasionally in math and in some computer languages, but it is not common.


Kappa: $\text{K},\, \kappa $: kă'pɘ. Commonly used for curvature.


Lambda: $\Lambda,\,\lambda$: lăm'dɘ. An eigenvalue of a matrix is typically denoted $\lambda $.  The $\lambda $-calculus is a language for expressing abstract programs, and that has stimulated the use of $\lambda$ to define anonymous functions. (But mathematicians usually use barred arrow notation for anonymous functions.)


Mu: $\text{M},\,\mu$: mū.  Common uses: to denote the mean of a distribution or a set of numbers, a measure, and the Möbius function. Don’t call it “u”. 


Nu: $\text{N},\,\nu$: nū.    Used occasionally in pure math,more commonly in physics (frequency or a type of neutrino).   The lowercase $\nu$ looks confusingly like the lowercase upsilon, $\upsilon$. Don't call it "v".


Xi: $\Xi,\,\xi$: zī, sī or ksē. Both the upper and the lower case are used occasionally in mathe­matics. I recommend the ksee pronunciation since it is unambiguous.


Omicron: $\text{O, o}$: ŏ'mĭcrŏn.  Not used since it looks just like the Roman letter.


Pi: $\Pi \text{,}\,\pi$: pī.  The upper case $\Pi $ is used for an indexed product.  The lower case $\pi $ is used for the ratio of the circumference of a circle to its diameter, and also commonly to denote a projection function or the function that counts primes.  See default.


Rho: $\text{P},\,\rho$: rō. The lower case $\rho$ is used in spherical coordinate systems.  Do not call it pee.


Sigma: $\Sigma,\,\sigma$: sĭg'mɘ. The upper case $\Sigma $ is used for indexed sums.  The lower case $\sigma$ (don't call it "oh") is used for the standard deviation and also for the sum-of-divisors function.


Tau: $\text{T},\,\tau$ or τ: tăoo (rhymes with "cow"). The lowercase $\tau$ is used to indicate torsion, although the torsion tensor seems usually to be denoted by $T$. There are several other functions named $\tau$ as well.


Upsilon: $\Upsilon ,\,\upsilon$  ŭp'sĭlŏn. (Note: I have never heard anyone pronounce this letter, and various dictionaries suggest a ridiculous number of different pronunciations.) Rarely used in math; there are references in the Handbook.


Phi: $\Phi ,\,\phi$ or $\varphi$: fē or fī. Used for the totient function, for the “golden ratio” $\frac{1+\sqrt{5}}{2}$ (see default) and also commonly used to denote an angle.  Historically, $\phi$ is not the same as the notation $\varnothing$ for the empty set, but many mathematicians use it that way anyway, sometimes even calling the empty set “fee” or “fie”. 


Chi: $\text{X},\,\chi$: kī.  (Note that capital chi looks like $\text{X}$ and capital xi looks like $\Xi$.) Used for the ${{\chi }^{2}}$distribution in statistics, and for various math objects whose name start with “ch” (the usual transliteration of $\chi$) such as “characteristic” and “chromatic”.


Psi: $\Psi, \,\psi$; sē or sī. A few of us pronounce it as psē or psī to distinguish it from $\xi$.  $\psi$, like $\phi$, is often used to denote an angle.


Omega: $\Omega ,\,\omega$: ōmā'gɘ. $\Omega$ is often used as the name of a domain in $\mathbb{R}^n$. The set of natural numbers with the usual ordering is commonly denoted by $\omega$. Both forms have many other uses in advanced math.  

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Computable algebraic expressions in tree form

Invisible algebra

  1. An  expression such as $4(x-2)=6$ has an invisible abstract structure.  In this simple case it is

using the style of presenting trees used in academic computing science.  The parentheses are a clue to the structure; omitting them results in  $4x-2=6$, which has the different structure

By the time students take calculus they supposedly have learned to perceive and work with this invisible structure, but many of them still struggle with it.  They have a lot of trouble with more complex expressions, but even something like $\sin x + y$ gives some of them trouble.

Make the invisible visible

The tree expression makes the invisible structure explicit. Some math educators such as Jason Dyer and Bret Victor have experimented with the idea of students working directly with a structured form of an algebraic expression, including making the structured form interactive.

How could the tree structure be used to help struggling algebra students?

1) If they are learning on the computer, the program could provide the tree structure at the push of a button. Lessons could be designed to present algebraic expressions that look similar but have different structure.

2) You could point out things such as:

a) “inside the parentheses pushes it lower in the tree”
b) “lower in the tree means it is calculated earlier”

3) More radically, you could teach algebra directly using the tree structure, with the intention of introducing the expression-as-a-string form later.  This is analogous to the use of the initial teaching alphabet for beginners at reading, and also the use of shape notes to teach sight reading of music for singing.  Both of these methods have been shown to help beginners, but the ITA didn’t catch on and although lots of people still sing from shape notes (See Note 1) they are not as far as I know used for teaching in school.

4) You could produce an interactive form of the structure tree that the student could use to find the value or solve the equation.  But that needs a section to itself.

Interactive trees

When I discovered the TreeForm command in Mathematica (which I used to make the trees above), I was inspired to use it and the Manipulate command to make the tree interactive.


This is a screenshot of what Mathematica shows you.  When this is running in Mathematica, moving the slide back and forth causes the dependent values in the tree also change, and when you slide to 3.5, the slot corresponding to $ 4(x-2)$ becomes 6 and the slot over “Equals” becomes “True”:

As seen in this post, these are just screen shots that you can’t manipulate.  The Mathematica notebook Expressions.nb gives the code for this and lets you experiment with it.  If you don’t have Mathematica available to you, you can still manipulate the tree with the slider if you download the CDF form of the notebook and open it in Mathematica CDF Player, which is available free here.  The abstractmath website has other notebooks you may want to look at as well.

Moving the slider back and forth constitutes finding the correct value of x by experiment.  This is a peculiar form of bottom-up evaluation.   With an expression whose root node is a value rather than an equation, wiggling the slider constitutes calculating various values with all the intermediate steps shown as you move it.  Bret Victor s blog shows a similar system, though not showing the tree.

Another way to use the tree is to arrange to show it with the calculated values blank.  (The constants and the labels showing the operation would remain.)   The student could start at the top blank space (over Times)  and put in the required value, which would obviously have to be 6 to make the space over Equals change to “True”.  Then the blank space over Plus would have to be 1.5 in order to make multiplying it by 4 be 6.  Then the bottom left blank space would have to be 3.5 to make it equal to 1.5 when -2 is added.  This is top down evaluation.

You could have the student enter these numbers in the blank spaces on the computer or print out the tree with blank spaces and have them do it with a pencil.  Jason Dyer’s blog has examples.

Implementation

My example code in the notebook is a kludge.  If you defined a  special VertexRenderingFunction for TreeForm in Mathematica, you could create a function that would turn any algebraic expression into a manipulatable tree with a slider like the one above (or one with blank spaces to be filled in).  [Note 2]. I expect I will work on that some time soon but my main desire in this series of blog posts is to through out ideas with some Mathematica code attached that others might want to develop further. You are free to reuse all the Mathematica code and all my blog posts under the Creative Commons Attribution – ShareAlike 3.0 License.  I would like to encourage this kind of open-source behavior.

Notes

1. Including me every Tuesday at 5:30 pm in Minneapolis (commercial).

2. There is a problem with Equals.  In the hacked example above I set the increment the value jumps by when the slider is moved to 0.1, so that the correct value 3.5 occurs when you slide.  If you had an equation with an irrational root this would not work.  One thing that should work is to introduce a fuzzy form of Equals with the slide-increment smaller that the latitude allowed in the fuzzy Equals.

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