Thursday, December 29, 2005

My thoughts on "Math is hard" - the myth

Before I begin, my blog mama has posted her thoughts on this here. If you haven't read them, go take a look.

I don't think that math is hard even though there are some maths which totally befuddle me (probability being the best example). However math, like most disciplines, requires building a foundation before erecting the house. I would like to write a proof for one of those interesting little factoids that arise in math. I don't know if it's feasible and I don't care. This is a fun mental exercise for me. But before I can even begin, I have to learn some additional math - my foundation isn't complete yet. It's a very good thing that I enjoy math.

Now getting to the title of this post. Earlier this month I had a short email discussion about the myth of mathematics. Actually, it wasn't that short and I'm going to copy the bulk of what I had to say here. But rest assured, there is no actual math in these posts - just my ideas and opinions.

The email exchange started with a request that I explain what it was that I called the myth of mathematics. This was my somewhat long winded reply.

I started thinking about this when I took the Praxis and some education classes. (I wasn't sure if I wanted to teach.) On average non-mathematics majors who were planning on teaching took the math section of the Praxis at least twice before passing. Many took it three or more times. I think the most retests I heard about was five. I wasn't overly concerned about the students who were planning on teaching English or History or anything that wasn't math or science, but virtually all of the students preparing to teach on the elementary level needed several tries before they passed. At the same time I kept on hearing "I can't do math, I'm bad at math, math is so hard." Well maybe I'd buy that differential equations are hard, but the Praxis math section is almost all arithmetic! So here are all these future elementary teachers who will have to teach arithmetic telling me that they can't do it, hate it, think it's hard, and so forth. And they're proud of it and see nothing wrong with repeating a basic test multiple times. At the same time, students who needed to retake the reading and writing sections would barely admit to that. Not passing the reading and writing section was socially unacceptable even for the math majors.

This is the mythology of mathematics - the belief that math is difficult and can only be done by "special" people. Math people are the priests in some esoteric religion that the ordinary man (or woman) is barred from. This is crap.

Before you can reveal the falsity of this myth, you need to understand that mathematics is not arithmetic. Just as you need to learn to count before you learn to add and subtract, you need to learn arithmetic before basic mathematics are accessible. Counting is a universal skill - baby's have a sense of number and basic counting skills before they are verbal; many animals also exhibit an awareness of numerosity and number sense. Baby's understand that if you put two stuffed toys behind a screen, when the screen is removed there should be two toys not one or three. This means that children are primed by biology to count, to add and to subtract before they ever enter a classroom.

Then school begins. The teacher honestly believes that arithmetic is mathematics and that math is hard. The teacher probably also dislikes and maybe fears arithmetic/math. Even if the teacher never says that math is hard and gives no outward indication of dislike or fear, children are almost eerily aware of the underlying feelings. So with no intention of perpetuating the myth, the teacher does. And when the student goes home, they will more likely than not see these same feelings in their parents.

Interestingly enough, through the fourth or fifth grade students say that they like math. There seems to be a delayed reaction. This may be a response to the mathematics standards that teachers are expected to meet. As the standards demand a more technical background, the ability to meet the standards becomes harder to reach. State standards are usually based on the standards set my the National Council of Teachers of Mathematics (NCTM). See this web page for an overview of the standards - www.nctm.org/standards/gradeband.htm. The problem? The teachers have no idea what the NCTM is talking about since the standards are written by and for mathematics educators. NCTM does state that mathematics should be taught by math professionals, but how many school systems can actually afford (and justify) the additional staff for this one subject?

To add insult to injury, the state of New Hampshire standards expect elementary teachers to cover the following math topics in the elementary grades - Problem Solving and Reasoning, Communication and Connections, Numbers, Numeration, Operations, and Number Theory, Geometry, Measurement, and Trigonometry, Data Analysis, Statistics, and Probability, Functions, Relations, and Algebra, Mathematics of Change, Discrete Mathematics (see www.nheon.org/frameworks/matrix.php for information).

So the little aliens get out of elementary school and have, usually for the first time, a math teacher. The math teacher has a vested interest in maintaining the idea that math is special and not every student is going to be able to do math. Again, this is crap. Every student has the same evolutionary history and so has the same cognitive skills. (I'm ignoring special ed students, I know.) Some students will be better at accessing these skills, but this is no different than any other skill. Some people can draw or write or play baseball (and so forth) better than others. Math is the same.

Once we learn that the cognitive skills used in math have been in play since early man began using tools, educators can start teaching to these innate skills. Math is no longer a secret religion, but a tool and the myth that "math is hard" should evaporate.


I was then asked where the myth originated and answered as follows:

Origins of the math myth are rooted in the historical development of mathematics within western culture and society. The earliest artifacts of western mathematics appear in the mideast in Mesopotamia and then later in Egypt. In both cases, mathematics was the purview of the priests and scribes. Although the math itself was not simple, its needs were - tax collection and calculating the movement of the heavens for predictions. In the cultures and society of that time, the people learning and doing math were largely from the priestly classes and male. Calculating for commerce was a matter of keeping a tally - simple counting and arithmetic. As mathematics emigrated into Greece, the mathematicians were still primarily male and upper class.

It is also of note that mathematics was defined as geometry. Algebra arrived in the west around 500 AD. Muslim scholars had developed algebra for use in trade and tax collection. By this time western culture was pretty thoroughly male dominated and the educational institutions and educators were to a large extent tied to the church. Although there were female mathematicians, the prevailing societal rules did not encourage women to learn or practice mathematics. So math was developed and used first by priests who were male. Later use was pretty much restricted to males who were members of the wealthy ruling classes. Since the practicioners of math were deemed special in some way, mathemathics acquired a reputation which grew into myth.

Reading, by the way, does not suffer from this elitism. I think because learning to read the Bible would have been considered "good" in some sense.

Notice that I've said western mathematics.

Since we all engage in symbolic reasoning via language everyday, that should not make mathematics unaccessible. Although I question the theory that mathematics is entirely an adaptation of language based brain function, a good portion of its outward character is.

The final question was on my definition of cognition and cognitive evolution. So for your edification, here's the last of my weird ramblings.

The point of the research I want to do in graduate school is to understand the cognitive processes underlying math and then find ways to use this information to develop a better way of teaching math. At this point, I'm just a tad light on specifics partially because (with one exception) there are no math cog theories that cover actual mathematics rather than numbers and arithmetic.

The 19th century did see a radical change in mathematics. Virtually all mathematics prior to that time, including calculus, had been developed largely to fulfill practical needs (I know that I'm glossing over the few people like Euclid). All math was essentially applied math. By the middle of the 19th century a radical expansion of mathematics was in full swing. Topology, set theory, non-euclidean geometry, multi-dimensional maths, ring and field theories in algebra, etc. were created by the mathematicians within a very short period of time. This was to me the birth of "pure" mathematics - math for math's sake with no need for application to real world problems. The fact that these new maths found applications later on is nice, but wasn't the point behind their development.

I have a theory - one that I'd like to look at further later on. I believe that human cognition went through an evolutionary change. This change was the result not of biologic pressures but cultural pressures. Cognition has evolved over time. When early hominids started using stone tools, there was a cognitive change. Another occurred when tools changed from one all purpose tool to many specific tools. Yet another when early humans began to bury their dead. The appearance of body decoration and the later appearance of cave art and 3-dimensional art are also indicative of evolving cognitive processes. Each of these changes took thousands of years to develop and, I think, were driven by a Darwinian style evolutionary process.

The emergence of new mathematics in the 19th century was too sudden to be the result of this type of evolutionary process so culture must have been the driving force. (Language, by the way, evolves through cultural/societal pressures and does so at a fairly rapid pace.) Understanding this change may be one of the clues to understanding how we "do" math. I haven't decided yet, but it's fun to think about.





Wednesday, December 28, 2005

Math is hard?

The latest post at Ars Mathematica is very simple - "Math is hard. Discuss." The comments from mathematicians and math educators while few so far are very interesting. If you have a burning desire to see what mathematicians think of math and math education, go read the comments.

After you've read the comments, come back here and tell me what you think. Is math hard? Why?

Sunday, December 25, 2005

Greetings and Solicitations

A while back I worked at a company that I referred to as the UN of companies. Each year the department I worked in had a holiday lunch with the employees bringing in food from their respective countries. The last year I was there the only day when most of the people could get together happened to be the first day of Hanukkah. One of the employees wrote "Happy Hanukkah" on the white board after much discussion on the correct spelling. Another added "Merry Christmas." Then everyone started adding greetings for their particular holidays. So there we were eating really great food and adding holiday greetings until the board was covered with as many different religious greetings as there were different people. This always struck me as the perfect Christmas meal - Christians, Jews, Muslims, Buddhists, Taoist, Hindus, and probably others I've forgotten celebrating the holiday season and breaking bread together.

So in that spirit, I wish you all greetings and solicitations this holiday season.

Tuesday, December 20, 2005

Apparently disconnected ramblings

Can you find the connecting thread?

This morning I got to arrange payments for yesterday's heating oil delivery. Whoopee. So why have the gas prices come down since their spike after the hurricanes, but heating oil hasn't?

My grades are in. A little self-congratulation going on here and bragging
Course Section/TitleFinal GradeCredits
1 PSYC-495-01 Sem: Nat & Orig ConsciousA3.00
2 PSYC-340-01 Psychological TestingA3.00
3 PSYC-254-03 Research Methods LabA1.00
4 PSYC-252-02 Research Methods PsycA3.00


The Pennsylvania courts have thrown ID out of the science classroom. My faith in the intelligence of at least some people is restored. Part of the judge's 139 page opinion includes the statement that "We have concluded that it is not [science]" (from CNN) .

I learned a new word today. While on what already feels like the world's longest break from school, I'm going over the journal articles I pulled with information on previous studies relating to cognitive metaphor and simile. The basic idea is to chart out the methodologies used to find evidence for cognitive metaphors and determine how to use these methods when I design a test for my research project. I need a test that will show evidence of unconscious use of a containment metaphor. Supposedly we all have this innate metaphor (I have a study that showed its existence in children between 9 and 19 months). According to one theory of mathematical cognition, this is a grounding metaphor that we use when doing set theory.

And the new word is "telic" - (adj.) 1. gram. expressing end or purpose; 2. tending to a definite end. From Gk "telikos," pertaining to an end or cause. (from Random House Webster's Unabridged Dictionary, 2nd Edition. 2001)

Friday, December 16, 2005

School's out

New classes begin on January 17th. I already miss going to classes, but I have a small stack of books to read - "Red Lily" by Nora Roberts (3rd book in the Garden Trilogy), "Fermat's Enigma" by Simon Singh (from my cousin), "The City of Fallen Angels" by John Berendt, "Guns, Germs and Steel" by Jared Diamond (a friend sent me this one and the Berendt book), "Handbook of Mathematical Cognition" ed. by Jamie I.D. Campbell, and more chapters in "Analysis With an Introduction to Proof" by Steven R. Lay. Doesn't this sound like fun?

Sunday, December 11, 2005

What ever possessed me to...

choose a book that is heavy on philosophy for one of two critiques/analyses? I so don't care for philosophy. I've been trying to finish the ###### paper for a week and a half. At this point I'd like to finish the paper by saying that the rest of the book is crap. And I'm procrastinating.

Friday, December 09, 2005

Five Weird Habits

Teresa has tagged me with this meme although she asks if it should be the "5 habits of Weird People?" Perhaps it should be the 5 Hobits of the weirdly ordinary world. I don't know, but here you go.

1) the number one weirdest habit according to my friends is reading math text books for fun.

2) I was going to say that backing up my research files in 3 different places was weird, but that just may be anal.

3) after I eat a Dove dark chocolate I always fold the wrapper in thirds, then in thirds again, and then roll it up. Well sometimes I save them because I like the message.

4) I add up the numbers on license plates, but I have to find them in consecutive order (this semester I've gotten 10 thru 39).

5) I don't think I'll admit to any more, this is beginning to sound like I have a little personality problem.

Now the problem of tagging 5 people. I emailed Teresa and asked if I could arbitrarily pick five people from The Bad Example Family since I don't actually know 5 bloggers. Since she said it was okay, here they are:

_Jon of We Swear - cause the blog momma said you were just dying to answer this.

Sarah of That's Not Very Nice! because she tagged me before. - Well shoot, Sarah's already been tagged with this one. If anybody wanders by and wants to take her place, feel free.

VW Bug at One Happy Dog Speaks because my dog is one happy dog and he's speaking right. now.

Well that's three.

I guess for the fourth I'll see if I can tag Jenna (even thought she's not in the family) at Sorgin because she really needs more to do as the semester winds down.

Hmmmm. Who for a fifth?
Okay, how about Jeff at Ponytailed Conservative because I adore people who aren't at all PC.

Thursday, December 08, 2005

What do you want to be when you grow up?

One of the women I work with asked me this morning what I was going to be when I grow up. While checking my email tonight, the answer appeared. Now, back to the world's most boring paper. *grin*

I can't think of a title!

There used to be a bulletin board where I made lots of friends like Teresa my blog momma. One of the people there was a young woman who is now in her first year of college. She is awesomely (is that a word?) smart! In between completing various papers and getting ready for final exams, I've been reading her blog. And learning. Woohoo! Go check it out.

Sunday, December 04, 2005

Thursday, December 01, 2005

Things I've never wondered about

The latest post on Ars Mathematica starts with "If you’ve ever wondered what a spin glass was..." Now this is something I've never wondered about (if you have, follow the link to the Wikipedia entry) and, in my usual totally out of touch with reality fashion, I kept on thinking "spin the bottle" instead of "spin glass." The post was filed under "physics" so great confusion ensued - why would physicists be writing about spin the bottle? This meant that of course I had to follow the link to the referenced article - Spin Glasses for Pedestrians. After reading the abstract I've decided that it may be a very convoluted and wordy explanation of "spin the bottle" after all.

Tuesday, November 29, 2005

The bright spot in a really pissy day

I'm not going to go into the really pissy day part because I'd probably say something that would get me arrested. Just the bright spot.

I lost my wedding ring a while ago (during the flooding stuff). Now this really upset me. It was the ring my first husband gave me. The first year we were married we had one of those fights and I took off the ring and threw it across the room. Ed retrieved it, jammed it back on my finger and said (actually yelled), "promise me you'll never take that ring off again!" I did and I never have in the 25 years since, so losing it was upsetting. But in the course of trying to deal with the problem that started of the pissy day, I found my ring!

My world will be all right again.

Saturday, November 26, 2005

Knock, knock

Q: Whose there?
A: *silence*

From the San Jose Mercury News - Evolution site under fire

"Operators of a University of California-Berkeley Web site that is designed to help teachers teach evolution are being sued by a California couple who say the site improperly strays into religion."

The claim that the site strays into religion is based on the link to information provided here by the National Center for Science Education - Statements from Religious Organizations.

The rest of the story is here.

Rituals - mindless wandering because it's snowing

Every year on Thanksgiving morning I watch the Macy's parade. On Christmas Eve I usually go to a church service - candles, music, no sermonizing. New Year's Eve I watch the ball come down in Times Square. Shortly thereafter I have a birthday and whatever decorations are in the house are put away. This ritual has changed little over the years. There used to be trips to the grandparents and my godmother, but they're dead now so the visits are memories.

Rituals also exist in other ways. Our society and culture impose rituals on us. Wikipedia in its entry on ritual states that "rituals can have a more basic sociological function in expressing, inculcating and reinforcing the shared values and beliefs of a society." Weddings, graduations, and football games all include ritual.

We all have personal rituals that center on specific dates or holidays. Some of these grow out of our culture or belief systems. We also have rituals that are more private and hold an almost mystical quality such as repeating a series of actions before an exam or always wearing a particular outfit to job interviews. These type of rituals are designed by us to increase the probability of a hoped for result.

So I'm sitting here watching the snow fall and wondering if mathematics isn't one huge ritual.

Tuesday, November 22, 2005

Meme - my first

Sarah of That's Not Very Nice! tagged me with my first meme. I found myself having all sorts of weird reactions to this. Since most of the time I'm minimally social, I felt a little like the kid in school who is unexpectedly chosen first instead of last for a team. And then I started laughing because the first of the two class discussions I had/have to lead for the seminar was on an excerpt from Richard Dawkins 1976 book "The Selfish Gene". The title of the excerpt - Selfish Genes and Selfish Memes. This may include the first instance of the meme concept. Dawkins is writing about evolution and a new emerging replicator.


"...It is still in its infancy, still drifting clumsily about in its primeval soup, but already it is achieving evolutionary change at a rate which leaves the old gene panting far behind.

The new soup is the soup of human culture. We need a name for the new replicator, a noun which conveys the idea of a unit of cultural transmission, or a unit of imitation. "Mimeme" comes from a suitable Greek root, but I want a monosyllable that sounds a bit like "gene." I hope my classicist friends will forgive me if I abbreviate mimeme to meme."

Having bored you all with that, on to the meme itself.

1. Do you use an alarm clock to wake up in the morning?
I am so not a morning person, so what do you think? At least I haven't thrown the (expletive deleted) thing across the room lately and risked killing anything.

2. What time do you set it for?
At least an hour earlier than the latest time I can get up. There's some madness in this method.

3. Do you hit the snooze button, if so how many times?
You have to ask. Some days I beat it with my cat.

4. Have you ever abused an alarm clock?
Well let's see. When in college I threw one across the room, barely missed my way too cheerful morning person roommate, and watched it break against the concrete block walls. These days I use the remote control for the TV or the cat to beat the (expletive deleted) button into submission. (oooohhhh...this submission thing is just too too.)

5. It's time to spread some "Its Blogcess" linky love.

Rules of the game, as per Sarah:
First, copy and paste #1 - #5
(Make sure to link to: "It's Blogcess", which is the link in #5. Because it is always polite to link the one who started the linky love.)
Second, link to my site (because it's polite to link the site that tagged you)
Third, go and tag 5 other blogs, more if you like.
Fourth, Email the owner of, or post on the blogs that you have tagged, to inform them that you've tagged them.

Sarah changed the rules and picked only 3 others to harass with this. I'm following her lead and tagging:

My blog mama (forgive me?)
The Pixie who is temporarily off-line and way far away so she can't kill me
My friend Tal who may send the mole rangers after me for this.

Thursday, November 17, 2005

Who am I?

Today I learned that I am an outlier.

Honor's Research project is moving forward!

Yesterday I spoke with the third professor I wanted on my project committee and she said "yes"! Woohoo, the committee is complete. On to the next step - the formal project proposal.

Wednesday, November 16, 2005

Jokes, Math, and Sex

In the last few days my email has included a math joke from my blog momma and a math chain mail from a friend. Both somehow someway included sex. Yes, sex and math. Who would have thought? Since you all probably don't believe me, I'm including the joke and the chain mail.

THE JOKE

A professor of mathematics sent a fax to his wife. It read:

"Dear Wife:
You must realize that you are 54 years old, and I have certain needs which you are no longer able to satisfy. I am otherwise happy with you as wife, and I sincerely hope you will not be hurt or offended to learn that by the time you receive this letter, I will be at the Grand Hotel with my 18-year-old teaching assistant. I'll be home before midnight.-- Your Husband"

When he arrived at the hotel, there was a faxed letter waiting for him that read as follows:

"Dear Husband:
You, too, are 54 years old, and by the time you receive this letter, I will be at the Breakwater Hotel with the 18-year-old pool boy. Being the brilliant mathematician that you are, you can easily appreciate the fact that 18 goes into 54 many more times than 54 goes into 18. Don't wait up."

THE CHAIN MAIL

SEX is like Math: add the bed, subtract the clothes, divide the legs, and pray you don't multiply!
You have been bumped by the Blinky
Which means you are a hottie.
You will have good sex for 2 years if you send this to 6 to 9 people. If this is sent back to you, you know you are truly a hottie.

Saturday, November 12, 2005

Auctions

I love going to auctions. Sometimes I actually buy things at auctions. But I've never thought of auctions as subject to mathematical analysis. Once again, I've been proven wrong. LOL

Check out this post at Ars Mathematica. It's worth following the link to the Grimm survey and reading the pdf document.

Thursday, November 10, 2005

My new favorite math quote

I followed a link from Ars Mathematica to the site to download a book (Toposes, Triples, and Theories, by Michael Barr and Charles Wells). From there I followed the link, CWRU Mathematics Department Website to the Case Western Reserve University's Department of Mathematics.

And there I found a truly wonderful quote - "Mathematicians, when they work, engage in intensely serious play. They follow their curiosity into problems that interest them and toward the smell of a solution." Richard Preston, The New Yorker, April 11, 2005

I may have to add this to my repetoire.

Wednesday, November 09, 2005

Stopping in to visit my own life

I've been busily trying to get school work done. So far I have two short and one long paper in the works, one short paper done and turned in, one take home exam written up, and the data analysis done for an extra credit paper (I want a 100 not a 90 for a test grade). Somewhere in there I did some database work, went to see a local production of Godspell, did the normal work and school stuff, completed all the paperwork for the honors program, and got sick.

I have the yuckys right now. You can tell I do because I didn't go to work today and paid sick time isn't part of my job. So I'm feeling sorry for myself as I go through the process of notifying professors, work, and the students in a project group when I started laughing. Why? Because I remembered that my mother used to do some self-medicating with alcohol. On occasion she'd wake up with a hangover, but my mother never had a hangover. It was always something she ate or a stomach bug. So here I am with a list of symptoms that except for the chills sound a lot like a hangover and a little ticked off that I didn't even get to have the "fun" of having a drink or two. Sheesh

Tuesday, November 01, 2005

8th grade math?

I'm so proud of all of you budding mathematicians with your fabulous test results!

One of my friends emailed this link. Fortunately I passed. LOL







Congratulations, you got 10/10 correct!

Sunday, October 30, 2005

Retirement savings (a rant of sorts)

One of the various and sundry people who provide personal viewpoints on Sunday morning TV delivered an editorial this morning on the necessity of having personal retirement savings. Now I don't disagree with this and I'm pretty well resigned to working until I die because I no longer have any savings for retirement. Why is this? Well....

My late husband had early onset Alzheimer's disease (diagnosed in his early 50's). Neither his job nor mine provided any medical insurance let alone insurance to cover long term care. Once he could no longer be left home alone, I quit working to take care of him. The cost of home care was prohibitive - more than I was earning even with two jobs. His social security disability included Medicare, but medicare doesn't cover home health care, long term care, or prescriptions. Unfortunately he reached a point where I could no longer care for him so that both of us would be safe. Enter the long term care dilemma and using state Medicaid. This was the only way to provide him the necessary care, but excluding the house and including all monthly income, we were allowed....are you ready?.....$2500 in assests. By the time we took his social security income and the value of my car, we were at the limit. Note that any kind of insurance policy or prepaid burial was included in this assest limit. So I have no savings and will most likely not earn enough in the next ten years (theoretical retirement time) to save enough to even consider retirement. And by the way, I still have no medical insurance because I can not afford to pay for it with no access to a group policy. On the other hand I'm very healthy.

The sort of rant? We need to find a way around this kind of idiocy. I don't particularly want to lose half my income to taxes to cover health care and retirement, but the current situation is disastrous for many people. I'm lucky since I'm healthy, prefer working over sitting around getting old, and own my home. But what about the people who aren't as fortunate as I am? Are we going to end up with lots of homeless elderly people who have limited or no access to medical care? There are a lot of people who work hard all their lives, but are not in a position to save for retirement. They're called the working poor. Technically these days, I'm one of them.

That's all. And probably the only time I'll go political here. I much prefer math.

Mathematical "proof" of the existence of God?

Via Bayesian probability here

Tuesday, October 25, 2005

Is this a sign?

It's raining again - a nor' easter. There are flood watches and high wind warnings out. On my way to school and work this morning the radio played Creedence Clearwater's Bad Moon Rising. Do you think this might be a sign? *snicker*

Monday, October 24, 2005

Tidbits from Ars Mathematica

This past week my favorite math blog, Ars Mathematica, has had two posts that relate to the seminar (Nature & Origins of Consciousness) I'm taking this semester. I doubt the mathematician intended this, but stuff happens. So now I can entertain you with information you never wanted to know about.

On October 19th, this notice appeared: Week 222 of This Week’s Finds in Mathematical Physics is up. If you go to this link, #2 - #5, talk about the idea of singularity - not the black hole math singularity thing, but a point where technological development goes beyond us.
2) Charles Stross, Accelerando, Ace Books, New York. Also available at http://www.accelerando.org/book/

This is one of the few tales I've read that does a good job of fleshing out Verner Vinge's "Singularity" scenario, where the accelerating development of technology soars past human comprehension and undergoes a phase transition to a thoroughly different world. This is a real possibility, and it's been discussed a lot:

One of the things we've discussed fairly extensively in class is what constitutes consciousness and whether an AI machine or a robot is truly conscious. Since I'd never read about this particular idea (singularity), I followed the links to learn some more. Kind of scary when I'm reading that this point in technology is predicted to occur by 2030.

October 23rd's post says "According to a new paper on Arxiv, Goedel’s theorem is false. There you have it." I haven't read the paper, but I'm assuming that it refers to Goedel's incompleteness theorem. One of the more interesting books I've read on consciousness was written by a mathematician (Shadows of the Mind by Roger Penrose, 1994). In the book, Penrose uses Goedel's incompleteness theorem to refute the possibility of creating a conscious machine. Does this mean Penrose isn't buying the singularity idea?

Math and psychology meet on the horizon of physics.

Saturday, October 22, 2005

Counting and negative numbers question for all.

In mathematics there is a set of axioms called "Peano's Axioms" which are the basis for proof by induction. These axioms also build the set of natural numbers (1, 2, 3, .......) or counting numbers. There are some interesting studies by psychologists that seem to show that these axioms exist at an unconscious level in children and primates. The psychologists aren't referencing the axioms, it's just something I noticed when reading the papers. It's pretty cool and gives me something to think about when I'm driving.

So....here I am driving down the road and thinking about how we learn to count and when different number concepts were incorporated into arithmetic and mathematics. Which gets me to negative numbers. You can conceptualize the natural numbers and fractions (rational numbers), but how do you conceptualize a negative number? When numbers are thought of as quantities, what is a negative quantity? Any ideas out there?

If you have some uncontrollable desire to read about Peano's axioms, there is a good description here at Wikipedia.

Thursday, October 20, 2005

Happy Happy Joy Joy

My road is open in both directions! There's still paving and guard rails to go, but I can drive through the much much shorter way. I'm hoping the paving is done before the weekend since the local weather is forcasting heavy rains this weekend.

That's all.

Wednesday, October 12, 2005

Hypothesis, Theory, and Proof (very long)

One of the classes I'm taking this semester is a senior seminar entitled "Nature and Origins of Consciousness." We've looked at and discussed different historical explanations for consciouness and have now wandered into a discussion of intelligent design vs evolution. Which leads me to the concepts of hypothesis, theory, and proof (and a rant).

I'm finding that scientists, psychologists, mathematicians, and the general population use each of these terms in somewhat different ways. Everyone seems to use hypothesis to denote an unproved idea. A scientific or mathematical hypothesis appears to be a complete idea. It's the theory and proof concepts where problems occur.

For scientists and mathematicians, a theory (or theorem) is an hypothesis that has been proven. For psychologists, a theory is a logically organized set of statements that include definitions of various events or concepts, contain information about relationships between these events, explain the causes of the events where the hypothesis is simpler and more tentative. (I just had to memorize this for a test.)

Psychologists and scientists consider a theory proved when they can show significant evidence for it and when the theory exhibits good predictive validity. Einstein's theory of relativity was considered proven once phenomena predicted by the theory was shown to exist and when phenomena behaved in ways the theory predicted. In psychology proofs of theories are really statistical analyses of experimental data showing that the hypothesis based on the theory has a good probability of being true.

Since people (and the larger world in general) aren't particulary tidy, proof is not a 100% probability, true in all cases, concept for psychologists or scientists. This is also one of the reasons why theories are changed (or thrown out completely). As we learn more and gain more evidence, theories are modified. Sometimes the modifications become so unwieldy that science is forced into rethinking the whole idea. This is how science grows, but this does not automatically invalidate science.

Mathematicians (who are anal retentive) define a proof as a logical and rigourous mathematical argument. Unfortunately exhaustive evidence is not a proof. The mathematician must show that the hypothesis is true for all instances. For example, if I want to show that the sum of the numbers from 1 to n (n is any counting number greater than 1) = n(n+1)/2, just summing all the numbers from 1 to n=2, 1 to n=3, 1 to n=4, and so on for however long I want to go doesn't constitue proof. I could have done the sums a million times and all equaled n(n+1)/2, but that's not a proof. The proof involves an argument that shows this is true for all values of n. (BTW - this has been proved.)

The nice thing about this? Once a theorem is proved in mathematics, it stays proved. Euclid showed a proof of the PythagoreanTheorem that is still valid. The not so nice thing? Hypothesis that appear to work and for which there is a lot of evidence, but no proof. My personal favorite is the Riemann Hypothesis - unproven, but over a million solutions have been calculated and all agree with the hypothesis.

Remember what I said about people not being as tidy as numbers? I have to remind myself of this on a regular basis. Otherwise, I can't look at any psychological theory as proven. I learned the concept of proof in terms of mathematics.

Looking at the arguments "against" Darwin's theory of evolution, it seems that the general population defines both hypothesis and theory as an unproven idea and expects proof to be similar to a mathematicians - 100% true in all cases. The fact that evolutionary theory has significant amounts of evidence and excellent predictive value doesn't appear to convince people of its validity. I end up with a problem though when the counter proposal involves an unknown intelligence for which no evidence is available and a theory with no apparent predictive value.

Of course I also have a problem with the definition of a transcendental number - a number which is not algebraic. Hello, tell me what it is not what it isn't.

Mathematics and Arithmetic

Contagion asked, "What exactly is the difference between mathematics and arithmetic." There's a short answer and a longer answer. Since I live to ramble on about math, this is the longer answer.

Arithmetic is simply the calculation/computational discipline. In terms of my particular interest in mathematical cognition, I consider arithmetic one of the building blocks necessary to mathematics.

Mathworld (a really great online source for short answers to math questions and just fun reading) defines arithmetic as "the branch of mathematics dealing with integers or, more generally, numerical computation. Arithmetical operations include addition, congruence calculation, division, factorization, multiplication, power computation, root extraction, and subtraction. " (http://mathworld.wolfram.com/Arithmetic.html)

Balancing your checkbook is arithmetic. Having a degree in mathematics does not mean that I'm any better (probably significantly worse) at that than the average person.

Mathematics is more difficult for me to define because it covers many areas so back to Mathworld for a definition. "Mathematics is a broad-ranging field of study in which the properties and interactions of idealized objects are examined." (http://mathworld.wolfram.com/Mathematics.html) Things like logic, which is essential to proofs, figure prominently in the study of mathematics.

Mathworld's main index lists the following topics and serves as a fairly good example of what is included in mathematics: Algebra, Applied Mathematics, Calculus and Analysis, Discrete Mathematics, Foundations of Mathematics, Geometry, History and Terminology, Number Theory (my personal favorite), Probability and Statistics, Recreational Mathematics, and Topology. No arithmetic. The foundations of mathematics section includes axioms, logic, set theory, and theorem proving. Again, no arithmetic.

If you look at the history of mathematics, different maths were developed to serve the needs of people. All mathematics through calculus were application driven and strongly tied to arithmetical calculations. Early in the 19th century, math went crazy (probably too much math happy juice) and new maths were developed for their own sake. It happens that many of these "pure" maths have applications in today's world, but the applications came later and are only of interest to applied mathematicians (or scientists, engineers, etc.) using them.

Have I put you to sleep yet? LOL If not, here's one of my favorite math quotes - A mathematician is a machine for converting coffee into theorems.

Tuesday, October 11, 2005

The Bad Example Family has a black sheep

Check out my welcome -
It's a Girl

Reality Check

I got out to work and school today. One of my friends from school was evacuated early Sunday morning. She's been allowed back to her house to start cleaning up, but has lost quite a bit of stuff. Another student in one of my classes got caught in a flash flood on her way home from the library on Saturday. Her car, her text books, her notebooks, her computer, and her bike are toast. I know that I came out of this with little more than inconvenience.

If you want to see pictures and video from the area, check out the online coverage from the local TV station at www.theWMURChannel.com

And more rain (possibly 3 - 5") with flash flood warnings are up for Wednesday evening into Thursday.

Monday, October 10, 2005

It Rained

And then it rained some more. My road did not make out too well. If I make a right turn out of my driveway, this is what I find.

This is an issue for me because until it’s fixed my drive to work goes from 12.5 miles each way to 22.5 miles each way. Since I go from home to work to school, I’m ending up with way more driving than seems reasonable for a part-time job. Oh well. It’s just a temporary job and I’ll probably end up looking for something else soon since I’m in week 18 of a 6 week job. LOL


If I make a left turn and drive about 2 miles, this is what I find.

Around noon today, the town filled in this little hole enough for my housemate to get out and buy food for the birds (chickens, ducks, and geese – oh my!). But my road is now on the list of road closures on the TV, so who knows what tomorrow will bring.

On the bright side, my house is intact and dry.




Friday, October 07, 2005

Arithmetic and numerical processing are NOT Mathematics!

I recently purchased a relatively expensive book titled "Handbook of Mathematical Cognition" (2005). In the preface the editor defines mathematical cognition as "...the field of research concerned with the cognitive and neurological processes that underlie numerical and mathematical abilities." In the next paragraph, he says that the book is "a collection of twenty-seven essays by leading researchers in the field, and constitutes a comprehensive survey of state-of-the-art research on important facets of mathematical cognition."

Now, turn to the table of contents. Of the twenty-seven essays, eighteen are strictly number or arithmetic based papers. One definitely deals with mathematics. The remaining eight papers may or may not be about mathematics since I can not be sure just from the titles.

I had a number of reasons for buying the book and I'm not disappointed. Just from the research I've been doing the past two years, this was more or less what I expected.

I have a degree in Applied Mathematics which means that I spend way too much time explaining that I am not a good candidate for a job in accounts payable/receivable/payroll, etc. A math degree does not equal "good at and likes doing arithmetic." Arithmetic is one of the skills that provide a foundation for mathematics. It is not mathematics. Arithmetic and mathematics take place in different areas of the brain - they're neurologically different. So, everyone repeat ten times a day until you've learned it - arithmetic is NOT mathematics!
I'm Surprised

I didn't really expect anyone to wander off and read this let alone offer a welcome. Of course I haven't said anything yet that will cause you to enter a comatose state. Once I do, you all will remember Teresa's greeting (*** Sending over a bottle of Drambuie to start you off right - THAT should certainly be interesting - drunk blogging at its finest! *grin*) and beg me to hurry up and drink the Drambuie. The good news there - it only takes one drink and I'm toast. *snork*

Thanks to all of you who welcomed me to the family. *grin*

Thursday, October 06, 2005

Just because there should be more than one post.


Tentative classes for the spring semester
  1. Psychology of Learning
  2. History & Systems of Psyc
  3. Honors Seminar
  4. Honors Research Project
  5. Ethology (maybe)
Anybody who is foolish enough to ask what the research project is will be answered.
The stupid thing ate the first post. It was very important, so I will repeat.

This is here just so Snakeypoo can have a blog offspring who is a left of center math deviant geek.