The intersection of the two triangles is the cross-section of a cube, but in this post we wish to explore further the centre of similarity of the two triangles.

The line joining and satisfies

Similarly, the line joining and satisfies

Equating the two expressions gives and from which . The point of intersection is therefore at . By symmetry of this expression the line joining and also passes through this point. This is the point on the plane that is equi-distant from the xy-, yz- and xz- coordinate planes. It is also the central projection of the origin onto the plane along the vector parallel to .

In terms of the original two triangles this point is neither the centroid, incentre, orthocentre, circumcentre nor other commonly encountered triangle centre. Let us find the barycentric coordinates of this point (call it ) in terms of the triangle with vertices at .

The first barycentric coordinate will be the ratio of the area of to the area of . Since and have the same x-coordinate, this will be the ratio of the x-coordinates of to , which is . By symmetry it follows that the barycentric coordinates have the attractive form

Let the side lengths of be . Then by Pythagoras’ theorem, . Hence

.

By the cosine rule, (where ) which equals from the above expression. Therefore and similarly we obtain . Then

By the sine rule, ( being the circumradius of ) from which . Hence the barycentric coordinates of may be written in non-normalised form as

Comparing this with the coordinates of the orthocentre , the point is known as the square root of the orthocentre (see Theorem 1 of [1]). Note that the real existence of the point requires to be acute, which it is when . A geometric construction of the square root of a point is given in Section 8.1.2 of [2].

[1] Miklós Hoffmann, Paul Yiu. “Moving Central Axonometric Reference Systems”, Journal for Geometry and Graphics, Volume 9 (2005), No. 2, 127–134.

[2] Paul Yiu. “Introduction to the Geometry of the Triangle”. http://math.fau.edu/Yiu/GeometryNotes020402.pdf

I have been looking to write a series unpacking the *Standards for Mathematical Practice* (SMP) since reading a great text about Common Core Math & PLC’s edited by Timothy Kanold (see *Resources*); however, other topics have pulled me away from that goal. Recently though, Heinemann has charged some of their “mathematics heavyweights” with writing a series on the Standards, releasing one article for each SMP. So, with that as inspiration, I have been sharing & synthesizing the articles for my staff.

See below to read about Standards 1-4, and stay tuned for Part 2 in my series that will cover the Standards 5-8. Enjoy, and feel free to share some of your own bite-sized best practices in the comments section below!

**SMP #1: Make Sense of Problems & Persevering in Solving Them** – Problem solving is one of the hallmarks of mathematics (Kinold, et al.), and Sue O’Connell encourages us to consider the process (sequencing), strategies & attitude of our students as we teach them to solve problems. In planning for instruction we must provide *good *problems, which Kinold, et al. describe as having the characteristics listed below.

- Relevant to Students
- Meaningful Mathematics to Advance Understanding
- Opportunities to Apply/Extend Skills
- Support Multiple Strategies

Before we are ready to tackle the problem, our main focus must be on *comprehension *(Yes, I am putting my “literacy hat” back on). While we typically focus on *key words *(i.e. how many left), consider the role that close reading routines can play as students work. The CUBES framework, is one simple way to move beyond “key words” and can help reveal student understanding.

Kanold, et al. also provide a great framework to support students as they take “responsibility of organizing their thoughts about tackling the problem.”

**SMP #2: Reason Abstractly & Quantitatively**** – **In her blog entry, Pam Harris encourages us to promote quantitative reasoning by providing opportunities for students to *contextualize* & *decontextualize* numbers. Two important benefits to this reasoning is that students will be able to apply concepts in meaningful ways & (more importantly) reconstruct faded knowledge by reasoning with the content (Kanold, et al.). According to Kanold, et al., opportunities for students to develop number sense should include activities where students can:

- Express Interpretations about Numbers:
*Working with e**qual to/less than/greater than number sets, placed in context (i.e. pounds, length, etc.)* - Apply Relationships Between Numbers:
*Using the same numbers but in different contexts (as presented in Harris’ article)* - Recognize Magnitude of Numbers:
*Is 20 a big number? When eating cheeseburgers it is; when counting stars it is not. The discourse that accompanies this between teacher-student & student-student is essential!* - Compute:
*How many ways can you make the number 42?* - Make Decisions Involving Numbers:
*Is 15 minutes enough time to…? Where is the 5th house on the left?* - Solve Problems:
*Constructing/Deconstructing numeric operations culminates with problem solving activities. This is where explicit modeling should definitely take place & can be the focus of our mini-lessons. This can even be utilized as a “close reading” activity!*

**SMP #3: Constructing Viable Arguments & Critique the Reasoning of Others – **Steve Leinwand begins by stating that this standard contains the “nine most important words in the entire Common Core” and I couldn’t agree more! In addressing this standard, our students must move beyond simply solving the problem, to focusing on *how* the problem is solved. As Leinwand states, they must be encouraged to communicate, justify, critique & reflect. This will ensure that our focus shifts from a single approach, to “many approaches & justifications, thereby strengthening everyone’s learning.” In short, our classroom environment must provide opportunities for students to:

- Provide explanations & justifications as part of their solution processes (making & evaluating conjectures)
- Make sense of their classmates’ solutions by asking questions for clarification
- Facilitate meaningful discussions about mathematics

Click here to read my entry on “sentence stems” that help to support purposeful talk and here to see my take on student questioning & dialog (through the lens of Danielson’s FFT). Although our focus here is on mathematics, the scaffolding tools provided can surely be utilized across the curriculum.

**SMP #4: Model with Mathematics** – One of the things that is particularly interesting about SMP #4 (*Model with Mathematics*) is how explicitly it links to SMP #3 & SMP #5. In his article, Dan Meyer explains 5 essential pieces to SMP #4, which are outlined below.

- Identify Essential Variables:
*As we plan, it is essential that we provide REAL WORLD situations as a focus for student work (Meyer uses the register line, for example). This will help students to e**xtract the necessary information/facts that will allow them to solve the problem (for the literacy folks, this is “determining importance”).* - Formulate Models:
*When formulating models, we are talking about “nonlinguistic representations.” However, Kanold et al., reminds us that we must avoid focusing solely on manipulatives, as nonlinguistic representation also consists of diagrams, charts, graphs…and NUMBERS! See below for a nice planning framework that was included with the Kanold text.*

- Performing Operations:
*Here is where students will apply their math skills by synthesizing the essential variables & models to determine a conclusion.* - Interpreting Results & Validating Conclusions:
*Do the conclusions make sense…why/why not?*

Most importantly, Meyer reminds us that “modeling goes wrong when we do the most interesting parts for our students.” It is essential that we avoid “leaving them to focus solely on performing operations & interpreting results.” Instead, we must provide students the opportunity to “[make predictions], speculate which information matters, and decide what matters most.”

*Click on the graphic below to visit Heinemann’s site & **view each of the SMP articles that influenced this piece.*

You might not know it, but there has been a revolution in the human understanding of mathematics in the past 150 years that has undermined the belief that mathematics holds the key to absolute truth about the nature of the universe. Even as mathematical knowledge has increased, uncertainty has also increased, and different types of mathematics have been created that have different premises and are incompatible with each other. The *value* of mathematics remains clear. Mathematics increases our understanding, and science would not be possible without it. But the status of mathematics as a source of precise and infallible truth about reality is less clear.

For over 2000 years, the geometrical conclusions of the Greek mathematician Euclid were regarded as the most certain type of knowledge that could be obtained. Beginning with a small number of axioms, Euclid developed a system of geometry that was astonishing in breadth. The conclusions of Euclid’s geometry were regarded as absolutely certain, being derived from axioms that were “self-evident.” Indeed, if one begins with “self-evident” truths and derives conclusions from those truths in a logical and verifiable manner, then one’s conclusions must also be undoubtedly true.

However, in the nineteenth century, these truths were undermined by the discovery of new geometries based on different axioms — the so-called “non-Euclidean geometries.” The conclusions of geometry were no longer absolute, but relative to the axioms that one chose. This became something of a problem for the concept of mathematical “proof.” If one can build different systems of mathematics based on different axioms, then “proof” only means that one’s conclusions are derivable from one’s axioms, not that one’s conclusions are absolutely true.

If you peruse the literature of mathematics on the definition of “axiom,” you will see what I mean. Many authors include the traditional definition of an axiom as a “self-evident truth.” But others define an axiom as a “definition” or “assumption,” seemingly as an acceptable alternative to “self-evident truth.” Surely there is a big difference between an “assumption,” a “self-evident truth,” and a “definition,” no? This confusing medley of definitions of “axiom” is the result of the nineteenth century discovery of non-Euclidean geometries. The issue has not been fully cleared up by mathematicians, but the Wikipedia entry on “axiom” probably represents the consensus of most mathematicians, when it states: “No explicit view regarding the absolute truth of axioms is ever taken in the context of modern mathematics, as such a thing is considered to be irrelevant.” (!)

In reaction to the new uncertainty, mathematicians responded by searching for new foundations for mathematics, in the hopes of finding a set of axioms that would establish once and for all the certainty of mathematics. The “Foundations of Mathematics” movement, as it came to be called, ultimately failed. One of the leaders of the foundations movement, the great mathematician Bertrand Russell, declared late in life:

I wanted certainty in the kind of way in which people want religious faith. I thought that certainty is more likely to be found in mathematics than elsewhere. But I discovered that many mathematical demonstrations, which my teachers expected me to accept, were full of fallacies, and that, if certainty were indeed discoverable in mathematics, it would be in a new kind of mathematics, with more solid foundations than those that had hitherto been thought secure. But as the work proceeded, I was continually reminded of the fable about the elephant and the tortoise. Having constructed an elephant upon which the mathematical world could rest, I found the elephant tottering, and proceeded to construct a tortoise to keep the elephant from falling. But the tortoise was no more secure than the elephant, and after some twenty years of arduous toil, I came to the conclusion that there was nothing more that I could do in the way of making mathematical knowledge indubitable. (

The Autobiography of Bertrand Russell)

Today, there are a variety of mathematical systems based on a variety of assumptions, and no one yet has succeeded in reconciling all the systems into one, fundamental, true system of mathematics. In fact, you wouldn’t know it from high school math, but some topics in mathematics have led to sharp divisions and debates among mathematicians. And most of these debates have never really been resolved — mathematicians have simply grown to tolerate the existence of different mathematical systems in the same way that ancient pagans accepted the existence of multiple gods.

Some of the most contentious issues in mathematics have revolved around the concept of infinity. In the nineteenth century, the mathematician Georg Cantor developed a theory about different sizes of infinite sets, but his arguments immediately attracted criticism from fellow mathematicians and remain controversial to this day. The central problem is that measuring infinity, assigning a quantity to infinity, is inherently an endless process. Once you think you have measured infinity, you simply add a one to it, and you have something greater than infinity — which means your original infinity was not truly infinite. Henri Poincare, one of the greatest mathematicians in history, rejected Cantor’s theory, noting: “Actual infinity does not exist. What we call infinite is only the endless possibility of creating new objects no matter how many exist already.” Stephen Simpson, a mathematician at Pennsylvania University likewise argues “What truly infinite objects exist in the real world?” Objections to Cantor’s theory of infinity led to the emergence of new mathematical schools of thought such as finitism and intuitionism, which rejected the legitimacy of infinite mathematical objects.

Cantor focused his mental energies on concepts of the infinitely large, but another idea in mathematics was also controversial — that of the infinitely small, the “infinitesimal.” To give you an idea of how controversial the infinitesimal has been, I note that Cantor himself rejected the existence of infinitesimals! In Cantor’s view, the concept of something being infinitely small was inherently contradictory — if something is small, then it is inherently finite! And yet, infinitesimals have been used by mathematicians for hundreds of years. The infinitesimal was used by Leibniz in his version of calculus, and it is used today in the field of mathematics known as “non-standard analysis.” There is still no consensus among mathematicians today about the existence or legitimacy of infinitesimals, but infinitesimals, like imaginary numbers, seem to be useful in calculations, and as long as it works, mathematicians are willing to tolerate them, albeit not without some criticism.

The existence of different types of mathematical systems leads to some strange and contradictory answers to some of the simplest questions in mathematics. In school, you were probably taught that parallel lines never meet. That is true in Euclidean geometry, but not in hyperbolic geometry. In projective geometry, parallel lines meet at infinity!

Or consider the infinite decimal 0.9999 . . . Is this infinite decimal equal to 1? The common sense answer that students usually give is “of course not.” But most mathematicians argue that both numbers are equivalent! Their logic is as follows: in the system of “real numbers,” there is no number between 0.999. . . and 1. Therefore, if you subtract 0.999. . . from 1, the result is zero. And that means both numbers are the same!

However, in the system of numbers known as “hyperreals,” a system which includes infinitesimals, there exists an infinitesimal number between 0.999. . . and 1. So under this system, 0.999. . . and 1 are NOT the same! (A great explanation of this paradox is here.) So which system of numbers is the correct one? There is no consensus among mathematicians. But there is a great joke:

How many mathematicians does it take to screw in a light bulb?

0.999 . . .

The invention of computers has led to the creation of a new system of mathematics known as “floating point arithmetic.” This was necessary because, for all of their amazing capabilities, computers do not have enough memory or processing capability to precisely deal with all of the real numbers. To truly depict an infinite decimal, a computer would need an infinite amount of memory. So floating point arithmetic deals with this problem by using a degree of approximation.

One of the odd characteristics of the standard version of floating point arithmetic is that there is not one zero, but two zeros: a positive zero and a negative zero. What’s that you say? There’s no such thing as positive zero and negative zero? Well, not in the number system you were taught, but these numbers do exist in floating point arithmetic. And you can use them to divide by zero, which is something else I bet you thought you couldn’t do. One divided by positive zero equals positive infinity, while one divided by negative zero equals negative infinity!

What the history of mathematics indicates is that the world is not converging toward one, true system of mathematics, but creating multiple, incompatible systems of mathematics, each of which has its own logic. If you think of mathematics as a set of tools for understanding reality, rather than reality itself, this makes sense. You want a variety of tools to do different things. Sometimes you need a hammer, sometimes you need a socket wrench, sometimes you need a Phillips screwdriver, etc. The only true test of a tool is how useful it is — a single tool that tried to do everything would be unhelpful.

You probably didn’t know about most of the issues in mathematics I have just mentioned, because they are usually not taught, either at the elementary school level, the high school level, or even college. Mathematics education consists largely of being taught the right way to perform a calculation, and then doing a variety of these calculations over and over and over. . . .

But why is that? Why is mathematics education just about learning to calculate, and not discussing controversies? I can think of several reasons.

One reason may be that most people who go into mathematics tend to have a desire for greater certainty. They don’t like uncertainty and imprecise answers, so they learn math, avoid mathematical controversies or ignore them, and then teach students a mathematics without uncertainty. I recall my college mathematics instructor declaring to class one day that she went into mathematics precisely because it offered sure answers. My teacher certainly had that much in common with Bertrand Russell (quoted above).

Another reason surely is that there is a large element of indoctrination in education generally, and airing mathematical controversies among students might have the effect of undermining authority. It is true that students can discuss controversies in the social sciences and humanities, but that’s because we live in a democratic society in which there are a variety of views on social issues, and no one group has the power to impose a single view on the classroom. But even a democratic society is not interested in teaching controversies in mathematics — it’s interested in creating good workers for the economy. We need people who can make change, draw up a budget, and measure things, not people who challenge widely-accepted beliefs.

This utilitarian view of mathematics education seems to be universal, shared by democratic and totalitarian governments alike. Forcing students to perform endless calculations without allowing them to ask “why” is a great way to bore children and make them hate math, but at least they’ll be obedient citizens.

]]>When I spoke about something being interesting, what I had meant was that it’s something whose outcome I would like to know. In mathematical terms this “something whose outcome I would like to know” is often termed an `experiment’ to be performed or, even better, a `message’ that presumably I wil receive; and the outcome is the “information” of that experiment or message. And information is, in this context, something you do not know but would like to.

So the information content of a foregone conclusion is low, or at least very low, because you already know what the result is going to be, or are pretty close to knowing. The information content of something you can’t predict is high, because you would like to know it but there’s no (accurately) guessing what it might be.

This seems like a straightforward idea of what information should mean, and it’s a very fruitful one; the field of “information theory” and a great deal of modern communication theory is based on them. This doesn’t mean there aren’t some curious philosophical implications, though; for example, technically speaking, this seems to imply that anything you already know is by definition not information, and therefore learning something destroys the information it had. This seems impish, at least. Claude Shannon, who’s largely responsible for information theory as we now know it, was renowned for jokes; I recall a Time Life science-series book mentioning how he had built a complex-looking contraption which, turned on, would churn to life, make a hand poke out of its innards, and turn itself off, which makes me smile to imagine. Still, this definition of information is a useful one, so maybe I’m imagining a prank where there’s not one intended.

And something I hadn’t brought up, but which was hanging awkwardly loose, last time was: granted that the outcome of a single game might have an interest level, or an information content, of 1 unit, what’s the unit? If we have units of mass and length and temperature and spiciness of chili sauce, don’t we have a unit of how informative something is?

We have. If we measure how interesting something is — how much information there is in its result — using base-two logarithms the way we did last time, then the unit of information is a bit. That is the same bit that somehow goes into bytes, which go on your computer into kilobytes and megabytes and gigabytes, and onto your hard drive or USB stick as somehow slightly fewer gigabytes than the label on the box says. A bit is, in this sense, the amount of information it takes to distinguish between two equally likely outcomes. Whether that’s a piece of information in a computer’s memory, where a 0 or a 1 is a priori equally likely, or whether it’s the outcome of a basketball game between two evenly matched teams, it’s the same quantity of information to have.

So a March Madness-style tournament has an information content of 63 bits, if all you’re interested in is which teams win. You could communicate the outcome of the whole string of matches by indicating whether the “home” team wins or loses for each of the 63 distinct games. You could do it with 63 flashes of light, or a string of dots and dashes on a telegraph, or checked boxes on a largely empty piece of graphing paper, coins arranged tails-up or heads-up, or chunks of memory on a USB stick. We’re quantifying how much of the message is independent of the medium.

]]>capture feelings

can they express

the spheres

rotation

]]>So my best mate posted an article on his Instagram-themed blog *Instatute* recently, which was explaining how mathematics could be used to predict how many “likes” your next post will get.

I would like to go into greater detail on a few things, being the unnecessarily scientific and mathematical person I am.

The maximum lines and minimum lines of the number of likes against post number graph are (obviously) linear, with an equation in the form of y=mx+c. Subtracting the two would give you the size of the possible range of values your next post can take (something which I’ll be somewhat wrongly referring to as “disparity” from now on). One thing that can be observed is that disparity increases with post number. The uncertainty in the number of likes your next post will have increases with each post, which is definitely interesting.

One thing that can be deduced is that the larger the ratio of disparity to the number of likes received, the weaker the account, or rather, the less consistent. Think about it this way: a “weak” account will have inconsistent post quality and followers that are inconsistent with their likes to the account. By measuring this ratio, it is possible to obtain an indication of how “strong” or consistent an account and its followers are. I suggest the following equation.

Acc. Strength = 100% – [(0.5 x Disparity)/Likes On Latest Post] x 100%

Disparity is a linear function of post number, which can be obtained by subtracting the minimum function from the maximum function.

Using the equation, I was able to obtain an account strength of 76.4% for my account.

Meh.

*(Note: This equation assumes that the account used is set to Private, as private accounts have a natural rate of growth that generally follows a linear trend. Additionally, the following culture of private accounts are less subject to “trends” and “in-fashion” things.)*

*(Note 2: Don’t judge me.)*

not lacking the effort, still;

worlds were closing in,

feeling, there:

i’m here,

does not it shower down?

waves beneath these (sloped) realities,

foundations washed away,

like sand particulates,

without defined conglomerates;

engulfing static drones;

reigning on my head,

fractal openings.

goodbyes are not goodbyes,

when i’m dead.

waves dissolving through,

stone; entunneling.

failure wash away,

as she returns;

two engulfing one

with promises.

outstretched,

with pure intention;

with hopes of bridging gaps,

resonating, then.

blindness of eyes within,

three prepares itself

lessons to heal –

a hand upon a hand,

with one outstretched;

navigating paths,

as probing tentacles.

profundity awashes atrophy,

cells of shells

(trickle down her face)

as cells unlysed absorbing

rich as nourishment.

peering as a vortex peers,

through medium

fear afaces strength,

this latticework

beams of three, of one,

of nothing left but here.

phantasm overcomes,

semblance of the age,

ionic pouring streams,

and finding what is new

gnashing towards you;

unscreaming.

the wake of tides,

so clear; as mirrors now can see,

their unchained eyes,

as promises we keep.

———————

copyright 2015 all rights reserved

dedicated fondly to the grace of those hearts. ]]>

Curated by *Lefouque*

]]>

WHAT IS IT?

Mathematical reasoning is an essential part of literacy.

Literacy is “an individual’s ability to read, write and speak in English, and compute and solve problems at levels of proficiency necessary to function on the job and in society to achieve one’s goals, and develop one’s knowledge and potential.” – American National Literacy Act of 1991

HOW TO GET IT?

Mary Matthews of Blitz Games Studio, UK says ” Exploration, experimentation, team building, problem-solving and independent, personalized, differentiated experiences [will tap into] the full potential games can offer for learning.”

SOURCE:

]]>**Reasonable idea of Context **

School: This P-12 (split campus) context; locn Sunny Coast, urbanised, middle class, approx. 900 Primary students, average 25/class, diverse demographic with variety of cultural backgrounds. Well resourced across all areas inc ICT. Yr 3 classrooms have access to iPads within classroom block of 4 as well as computer lab world.

Students: Yr 4 class consists of 25 students 14g, 11b. Has one high function, low social skills ASD student, two low students. One student has been identified and G&T.

Staff: General TA support throughout various times of day, one-on-one TA support for ASD student

**Reasonable idea of Learning Objectives (WIP): **By the end of the Unit, students will be able to develop an increasingly sophisticated understanding of mathematical concepts and fluency with processes, and are able to pose and solve problems and reason in Statistics and Probability. They will be confident, creative users and communicators of mathematics, able to investigate, represent and interpret situations.

**Reasonable idea of criteria: **Statistics and Probability/Chance (Construct)

Conduct chance experiments, identify and describe possible outcomes and recognise variation in results (ACMSP067)

Data representation and interpretation (Transform)

Collect data, organise into categories and create displays using lists, tables, picture graphs and simple column graphs, with and without the use of digital technologies (ACMSP069)

**Possible Unit topic (WIP):**

Unleashing chance.

]]>Today at the MoSAIC exhibition at NC State University

]]>**Who Really Needs Cancer Surgery? Math Can Help Tell**

Endometrial cancer affects 48,000 US women per year. For patients with tumors greater than 2cm the effected organ(s) and lymph nodes may be surgically removed. Yet post-surgery analysis shows that 78% of these surgeries may have been unnecessary.

Mathukumalli Vidyasagar discusses how new computational algorithms from National Science Foundation-sponsored research have been successfully applied to cancer data in a clinical translational setting.’a0One such algorithm can predict the time to tumor recurrence in ovarian cancer patients and has been shown to predict the efficacy levels of several natural product compounds on lung cancer cells. This cutting-edge research in machine learning and computational algorithms has the potential to transform clinical practice and personalized cancer treatment therapies.

By: Live Science Videos.

]]>In this verse we read “For by him all things were created”. This verse tells us tells us the origin of math. It tells us God created all things. The word “all” includes everything, even math. How does math work? It works by God. In the next verse we read, “In Him all things hold together”. Math is precise, because God holds it all together. Mathematical truths are real only because they are not the product of the human mind, but of the divine mind. It was God who had invented math and it is He who has placed mathematical laws on the universe. For humanity, mathematics is a matter of discovery, rather than invention. As we learn the absolute consistency of math we know that an absolute consistent God is the inventor of that truth.

Everything in the universe involves mathematics. “All things” can be broken down into mathematical equations. The mountains, stars, oceans, planets; even the very air we breathe. The existence of math is in itself an act of creation. We know that the speed of light is 186,000 miles per second because we calculated it. But we did not create the equation, we discovered it. God invented it. From knowing the exact distance that the Earth had to be from the sun for life to exist and survive; to knowing how to construct human DNA to create a human being – we see evidence of the Master Mathematician throughout all creation.

God the Father is God, God the Son is God, and the God the Holy Spirit is God, and the three are one God. Mathematical proof of that is Infinity + Infinity + Infinity = Infinity.

Matthew 10:30 Even the very hairs of your head are all numbered.

Here is a fun little fun with numbers and then I will show you how to exceed 100% with God’s love in your life.

1 x 8 + 1 = 9

12 x 8 + 2 = 98

123 x 8 + 3 = 987

1234 x 8 + 4 = 9876

12345 x 8 + 5 = 98765

123456 x 8 + 6 = 987654

1234567 x 8 + 7 = 9876543

12345678 x 8 + 8 = 98765432

123456789 x 8 + 9 = 987654321

1 x 9 + 2 = 11

12 x 9 + 3 = 111

123 x 9 + 4 = 1111

1234 x 9 + 5 = 11111

12345 x 9 + 6 = 111111

123456 x 9 + 7 = 1111111

1234567 x 9 + 8 = 11111111

12345678 x 9 + 9 = 111111111

123456789 x 9 +10= 1111111111

1 x 1 = 1

11 x 11 = 121

111 x 111 = 12321

1111 x 1111 = 1234321

11111 x 11111 = 123454321

111111 x 111111 = 12345654321

1111111 x 1111111 = 1234567654321

11111111 x 11111111 = 123456787654321

111111111 x 111111111=12345678987654321

How far will knowledge, hard work, and attitude take you compared to the love of God? Let’s see. We will assign a number (1-26) to each letter of the alphabet (A-Z).

K-N-O-W-L-E-D-G-E = 11+14+15+23+12+5+4+7+5 = 96%

H-A-R-D-W-O-R-K = 8+1+18+4+23+15+18+11 = 98%

A-T-T-I-T-U-D-E = 1+20+20+9+20+21+4+5 = 100%

Nothing compares to the love of God which is able to do more than you could ever ask or want.

L-O-V-E-O-F-G-O-D = 12+15+22+5+15+6+7+15+4 = 101%

]]>**You’re Better At Math Than You Think**

Jere Confrey, an NSF-funded math educator at NC State University has found that differing pathways into math are available for different types of people.

By: Live Science Videos.

]]>So,with this Hebrew Shabbot(or Sabbath day) if those who actually open up their Bibles to read in a Jewish/Hebrew cycle of readings, & thoroughly discern, nonetheless studied either the Torah portion of **Tẓăw(v)-צַו** **, **from a Jewish/Hebrew mode of study for those out in the diaspora, living outside of Jerusalem or Israel. From the Jewish or Hebraic Calendar in the **5775** this Sabbath day would probably fall on the **7th-ז**, & the **8th- ח**, of the month of 13th month, which is a second month of [Nisan/]Abib

Readings:

Leviticus 6:8 – 8:36

Jeremiah 7: 21- 8:3

Jeremiah 9: 22-24

Hebrews 7:23- 8:6

http://www.hebrew4christians.com/Holidays/Shabbat/Special/Zakhor/zakhor.html

Now, for in summation of the study we have in front of us; In this portion of the study of the Scriptures, Moses receives further instructions from God about the laws and statutes for the “Sacrifices”/”Offerings,” that are to be administered by the preiesthood for the children of Israel.

http://www.chabad.org/holidays/purim/article_cdo/aid/644313/jewish/Zachor.htm

This portion of the study of the Scriptures, deals with the God’s instruction to Moses, in regards to the sacrifices(*Korbanot*-**קורבנות**/*Q’werban*-**ቍርባን**), in categorization. Moses, addressed Aaron & his sons, [the priesthood] in accordance to the ordinances of the sacrifices. Aaron & his descendants were to dress themselves in the garments of linen to administer the works within the tabernacle.

The offerings were to be prepared with flour, oil, frankincense and to be burnt upon the altar to God. What was left over was to be eaten by the priesthood with unleavened bread, in the **Holy Place** of the *Mishkan*(Tabernacle).

[Lev. 6:14-23]

After the laws were laid for the foundations of the sacrifices/offerings, Moses was to consecrate his brother Aaron, and his sons(*along with ALL his descendants*) to be the anointed priesthood, to carry out the works of God for the children of Israel.

Moses, and his brother Aaron, carried out all of the instructions given to them by God. In the wilderness they erected a Tabernacle, to the God that brought them out of bondage with the intent of bringing them to a land set forth for them to inherit. Aaron’s sons were designated to perform the services of the Tabernacle upkeep, for the children of Israel.

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1. Even Galileo didn’t know calculus, so why should I?

2. When I tried to reach my textbook, I could only reach half the distance, then when I tried reach it again, I could only reach half of that distance, then…

3. I know the proof, but there isn’t enough space to write it in this narrow margin.

4. I left it in the πth dimension, and I have to wait for another Pi day to be able to enter to that dimension again.

5. Since you were constantly talking about those imaginary stuff, I thought that the homework was just imaginary as well.

6. My math skill is at a transcendental level, so I didn’t bother to show you my homework, you won’t comprehend it anyway.

7. I accidentally dropped it on a paper shredder, and it shredded my homework into an infinite number of pieces.

8. My homework is about Godel’s Incompleteness Theorem, so I didn’t complete it as well.

9. I have a solar powered calculator and it was raining all day.

10. Wolfram Alpha was down last night.

11. I had a constant amount of homework. I tried to derive its purpose,

but I got nothing.

12. I tried to calculate the *π*th root of infinity and my computer exploded.

13. I was stuck in an infinite loop while using L’Hôpital’s rule.

14. I dug all around the yard but I never found the square roots that I needed

15. I ate too much pi and went to bed with a transcendental stomach ache.

16. I was unable to find some logs on our backyard.

17. I was exhausted from moving countless decimal points and raising complicated algebraic expressions to large powers.

18. The range of the improper integrals was between zero and infinity, so I was unable to finish them.

19. I put my homework in my bag but a four dimensional dog got in and ate it.

20. I tried to divide by zero and my homework burst into flames.

I hope that you would be able to determine the references that I used in some of the excuses especially in excuse #3. Well, if you can’t, you can always ask me.

]]>So, there I was. In the midst of that terrific traffic chaos in Pune. 6:00 p.m. People, tired from the day’s work in office, wanting to get home. Mothers, with two little ones on scooty peps. School kids chattering dime to dozen on their bicycles, barely noticing the smog and laughing delightedly in a way that only a “school-is-over” event can bring on. In the rickshaw next to my car was an old man looking agitatedly at his watch and urging the rickshaw wallah to hurry. Hope nothing’s wrong, I thought. Hawkers selling roses and kiddie toys looking shrewd and snappy about the traffic jam; an extra 4 minutes at the signal means brisk business. The signal turned green. And the traffic moved.

In different moods and colors, to different destinations from different start points, with a sense of happiness and with irritation, the traffic moved as *one*. One rhythm, one direction, one sense of purpose. Radio Mirchi squawked “Traffic moving slowly and *normally* through Karve Road”…Eh? What was that? Normally?

Why is traffic normal, when there’s nothing normal about the individuals who are its components? A gentle tap on the window and I saw those twinkling eyes giving me my answer. Oh my God, are you really George Polya? Yes, he nodded at me gently. And then said the sentence that absolutely put to rest any doubt I could ever have about his identity. “I thought I am not good enough for Physics and too good for Philosophy. Mathematics is somewhere in between.” As Polya settled down in my car, his usual spry animated self, the answers dawned on my mind. The Central Limit.

I mean, look at all around you. You find those large number of random, independent variables, each with their own distribution, each going through the motions of love, happiness, stresses and tensions. People. Random variables. And yet, the summation of the populace, life in general, moving normally. Wow, I breathed softly, hats off! Polya, you crazy genius, in which fit of madness did you coin the words “Central Limit Theorem” to explain the most fundamental theorem in probability statistics. Its not just probability statistics; time and again I connect to the Central Limit and the Law of Large Numbers as theorems that hold the explanation to the process of life itself.

As life passes by, as the days go by, we find that we all tend to an average way of life, some average thoughts, some average standard of living. Sure, each person defines his own average differently. But the fact is that as the days collect into our kitty, we start living the Law of Large numbers. On a particular day, life may treat us rather well or unkindly; but the average life of the average person is on an average…average! How “mean”, you may well say, but there you are! Such is life; defined by the Law of Large Numbers.

I like to think of life as a time series; mostly non-stationary. It rejects almost all attempts to stationarize it. Should you manage to, it mostly turns out to be an ARMA process, part Auto-Regressive (anthropologists love to call this genes), part made up of a Moving Averages of past errors (and the psychologists call this environmental adaptation). Predicting the future (how rum is it that the master of forecasting should be named Box; this is one guy who is definitely out-of-box!) requires an artistic, atheoretical and unique approach and mostly is riddled with crazy levels of failure. And yet, most of us, wizards to muggles, want to study and understand the mad art and science of forecasting. They call it divination, and we, astrology. *“Mars is bright tonight.”* We try to fit patterns to the lines on the forehead and assess the statistical probability of combinations of stars creating situations of a particular type for us. By the way, did you know that the Ordinary Least Squares as a method of minimizing errors was suggested by Gauss in his work on movement of celestial objects? There you go. All statisticians have been enamored by it. The movement of the stars. Their effect on our lives.

Gauss. The other genius. He is always around in my life too. The Gaussian wrackspurt, if you will….

The other day in college, end-of-the-session babble broke out in the classroom as I wound up the discussion. And suddenly a student asked me a question. Such was the babble that I couldn’t really hear him. And merely guessing what he was saying, tried to give a rational answer to his question. His look told me that I had inferred wrong. A gentle tap on the door .“*Inference. It always goes wrong when there’s way too much noise around you”*. I swung around to see him walking into my class. The babble was still on; but my mind had turned into a silent zone. There was total silence. Till he decided to talk. Gauss. The Greatest Mathematician since Antiquity.

“To hear the question and give the answer: That’s estimation, my dear. But, to interpret the question correctly. Ah, that is the art and science of inference. If the model errors are too vocal, exhibiting auto-correlation or heteroscedasticity noise, the inference on the estimates goes wrong.”

How did you know, my mind was asking. How does the solution come to you…Does it come in a flash of lightning, or is it a slow, prolonged process.. “I have had my results for a long time: but I do not yet know how I am to arrive at them.”

A good poem and a rainy day. Can’t get better than that. A poem titled “Yevgeny Onegin”by Pushkin. Hmmm, on the continuation of life even as death takes us over individually. And what’s this? Markov’s commentary on Onegin? Andrei Markov. Really? I mean, could this be THE A. Markov? As in, the Markov chains guy? What in the world does HE have to do with Pushkin now? Quite a lot, as the story goes.

We go back in time to meet Pushkin and Markov. In the tsarist Russia of late 1980s. At that time, statistics had developed to a point where probabilities of sequences of events were being estimated as the product of the two events happening independently. So, if a day can be sunny or rainy, the P(Day is sunny) = 0.5 and P(Day is sunny when earlier day was sunny) = 0.5*0.5 = 0.25. But, events are usually dependent, aren’t they? So, if today is sunny, doesn’t it impact the probability of tomorrow being sunny? Isn’t it true that probabilistic chains necessarily need an intertemporal dimension? The probability of an event today HAS to depend on the event occurring in the most recent past, a little lesser into more distant past and so on. And therein lies the genesis of Markov chains.

Markov was a man possessed in proving the interdependence of chains of events in a temporal sequence. His choice of a sequence to prove interdependence? The poem “Onegin”by Pushkin. Oh, this is sublime!

He chose the first 20,000 letters in the poem arranging them without punctuation or breaks, counting 8,638 vowels and 11,362 consonants in the process. There were 1104 vowel-vowel pairs, wherein a vowel follows another. Now, if occurrence of a vowel in the sequence is *independent* of what occurs earlier, then, the P(vowel-vowel pair occuring) = (8638/20000) * (8638/20000) = 0.19. That implies that in 20,000 letters, such combinations should have occurred 3731 times, nearly thrice the 1104 number of times that they occurred!

Wow, wow and more wow! This necessarily means that independence of vowels stands rejected! Thus, letters in a word are NOT independent you see; there is an overwhelming tendency for vowels to alternate with consonants in a language! The same logic has hence been used in applications as diverse as understanding motions of gas particles to creating the Google algorithm.

Onegin may be a poem par excellence; but it will go down in the annals of history as the literary masterpiece that served as a workhorse for a uber creative mathematician. How befitting that Onegin should talk about continuity; because if there is one theme that Markov explored, it was one of continuity and interdependence…A gentle tap on my shoulder…I need to go. It’s Markov.

]]>I found myself sitting around thinking, ‘this essay is pretty good’ – it got a pretty good mark from my philosophy lecturer. So I thought: ‘what the heck, I’ll post it on the blog’. While it is a little outside the traditional area of religion which most of you probably come to this blog for, my tag line does make reference to ‘logic’ and ‘reason’ – somethings I hope you will find a lot of in my essay. Perhaps the question of logicism is interesting to some of you; it is essentially the project of reducing mathematics to logic. For me, I picked the question because I always wanted to know about mathematics: we use it everyday, but we hardly ever think that hard about it. So I did some hard thinking, and this essay was the result. If you are interested in analytic philosophy, Bertrand Russell or the nature of mathematics, then this essay is for you. If not, try read it anyway: philosophy never hurt anyone (don’t quote me on that).

**The key thesis of Logicism is that mathematics can be reduced to logic. What does it mean to reduce mathematics to logic? And what is the philosophical point of the exercise?**

**The Facetious Question**

Thinking about mathematics is hard. It is hard because we are so familiar with it; basic arithmetic and the ability to count seem almost ingrained in the human experience. Just ask a young child her age, she might say something like “I am three and a half”. The fraction demonstrates an awareness on the part of the child that one day she will be 4, the successor of 3. Given our intimate relationship with mathematics from the youngest age, it can seem almost facetious to inquire into the truth that 1 is followed by 2, or 1+1=2. But philosophers have a penchant for seemingly facetious questions; Bertrand Russell expended a lot of energy considering the truth of mathematical propositions. Russell spent a lot of time trying to reduce mathematics to logic – I will firstly explore what that means. Then I will consider the impetus for this logicist project, considering what is at stake if the logicist project cannot succeed. As I soon hope to demonstrate, in the world of philosophy ‘why does 1+1=2?’ is not a facetious question.

**What does it mean to reduce mathematics to logic?**

One often hears from lay people that mathematics is logic. What the lay person often does not appreciate is that this proposition is not self-evident or obvious – this claim is the result of serious philosophical work. We are greatly indebted to the philosophical efforts of men like Frege, Peano and Russell who all played parts in the logicist effort to weave the threads of mathematics and logic into one seamless cloth. They tried to achieve this by reducing mathematics to logic.

In order to understand what this means, let us try to understand how the logicist seeks to relate mathematics to logic. In the opening line of Russell’s Introduction to Mathematic Philosophy, he states that “Mathematics is a study which, when we start from its most familiar portions, may be pursued in either of two opposite directions” (1). One can move constructively towards ever greater complexity, e.g. differentiation, integration etc. Or one can move towards greater abstractness, towards simplicity perhaps. In the words of Russell “we ask instead what more general ideas and principles can be found, in terms of which what was our starting-point can be defined or deduced” (1).

This abstract idea could benefit from an elucidatory metaphor. Imagine you have a set of basic tools with which you can do basic things. Those basic tools can be used to build more tools, which can create ever more complex constructions. But how are the basic tools built? Perhaps from a simpler set of tools, themselves formed by the primordial tool: human hands. For someone like Russell, human hands are logic, basic tools arithmetic, and the more complex tools higher mathematics like differentiation. The task of reducing mathematics to logic is similar to showing how the most complex particle accelerator was made from a screwdriver, itself the product of human hands. While a little clumsy, this metaphor should aid one in visualising the logicist project.

Keeping in mind the earlier metaphor, let us now equate tool with theory. Helpfully Soames provides us with the language and the categories to unpack the features of a theory (133). A theory begins with axioms: propositions which are accepted without proof, taken as given. These axioms contain the primitive vocabulary of the theory which is used to express concepts and relations that we know without definition. Definitions can also be added which are terms that do not appear in the axioms, but are nevertheless useful to have. Theorems constitute the statements which proceed logically from the axioms. In summary, Soames states “In general, we will take a theory to involve a set of axioms that are accepted without proof, plus, in some cases, a set of stipulative definitions that define new terminology in terms of the theory’s primitive vocabulary. Theorems are logical consequences of the axioms plus the definitions. (134)”

Now we are equipped with the language to understand what it means to reduce mathematics to logic. Assume we have two theories, T1 being logic and T2 being mathematics. To reduce T2 to T1 means to describe the axioms of T2 with the primitive vocabulary and definitions of T1. The end result according to Soames is that “theory 2 comes to be seen as an elaboration of what was already implicitly present in T1” (134). Harking back to the metaphor, the result is to show that human hands built the particle accelerator because the complex tools that created the accelerator were themselves the products of human hands; human hands built it all along.

In history, there were several important steps, or reductions rather, that Russell was indebted to. Chief among them was the arithmetization of natural numbers (Ongley and Carey, 25). The ancient Greeks separated geometry and arithmetic due to the problem of irrational numbers (these are numbers which cannot be expressed by ratios of natural numbers e.g. π). The work of Descartes in the 1600’s, and then of Dedekind in the 1870’s helped to marry irrational numbers back to rational numbers in his theory of real numbers. Real numbers (T2) then came to be expressed in terms of natural numbers (T1). The final step was Peano’s axioms which showed that the “entire theory of the natural numbers could be derived from three primitive ideas and five primitive propositions in addition to those of pure logic” (Russell, 5).

The long process of arithmetization produced an interlocking set of reductions where all the disparate branches of mathematics could eventually come to be understood in the vocabulary of Peano’s axioms. The incompleteness theory of Gödel would eventually demonstrate that no set of logical axioms could capture the entirety of arithmetical truth, but at the time Russell was doing his work, it was thought that Peano’s axioms were complete (Soames, 157). Peano posited five axioms, which used three primitive concepts: zero, successor, and number. The program of logicism sought to express Peano’s axioms in terms of logical concepts, and derive Peano’s five axioms from logical truths (Ongely and Carey, 30). Russell set out to do this by manipulating his logical definition of number (known as the Frege-Russell definition). Russell defined number as “… anything which is the number of some class” (19). From this logical expression of number, Russell was able to express the rest of Peano’s axioms using logic – the details of this forming the subject matter of his Introduction to Mathematical Philosophy.

In the end, the logicist program failed, Gödel’s proof dealing the coup de grâce. According to his proof, some arithmetical truths must remain unproven; therefore, the entirety of arithmetic cannot be captured by logic. Nevertheless, with this brief review in mind, we are now in a good place to review what it means to reduce mathematics to logic. Theoretical reduction is the feat of expressing the primitive vocabulary of T2, in the language of T1, and deriving the axioms of T2 from the axioms of T1. Peano’s axioms (T2) constituted the final reduction within mathematics. Russell took the next step in reducing Peano’s theory into logic (T1). In a sense reducing mathematics to logic confirms they are one and the same, as Russell states: “They differ as boy and man: logic is the youth of mathematics and mathematics is the manhood of logic” (194).

** Why would one want to reduce mathematics to logic?**

Having considered what it means to reduce mathematics to logic, let us now return to the question ‘why does 1+1=2?’ To many, the question may seem facetious: the truth is self-evident; the proposition is the bed rock that cannot further be challenged. In the realm of philosophy however it is a very serious question with high stakes.

The impetus for the logicist program can find its roots generally in the spirit of the analytic movement. The early shakers and movers of analytic philosophy, philosophers like Russell and G E Moore, were rallying against the absolute idealism which reigned supreme in philosophy towards the end of the nineteenth century. Schwartz writes that “Analytic philosophers got their inspiration, ideas, problems and methods from British empiricism, formal logic, mathematics and natural science” (2). Understanding the concerns of empiricism, in contradistinction to rationalism can help us understand the impetus behind logicism. The rationalist posits that some of our knowledge, for example mathematics and geometric notions are innate in the mind, or come from pure reason. The empiricist however is devoted to the idea that knowledge is derived from experience (Schwartz, 39). The project of logicism seeks to account for the knowledge of mathematics in a way amenable to empiricism.

Prior to the work of Russell, in his own words “Geometry, throughout the 17th and 18th centuries, remained, in the war against empiricism, an impregnable fortress of the idealists” (quoted in Schwartz, 13). Empiricism simply had trouble accounting for mathematics. There is no way that our experience of the world could account for the infinitude of prime numbers (Schwartz, 14). Further, no test or observation could be conceived to show that 1+1 does not equal 2; and if this is the case, then our knowledge of this cannot be derived from experience. We know mathematical propositions to be true, independent of our experience, which makes mathematical knowledge a priori.

The a priori nature of mathematical knowledge was a point in favour of the rationalist. With no empirical account, the rationalist might posit pure reason as the source of that knowledge. In establishing that claim, they might make further appeals to innate ideas or pure reason in understanding other things such as metaphysics: development surely anathema to the strict empiricist. Logicism however provided a means to bring the truths of mathematics within an empiricist framework.

Recall that the aim of logicism is to reduce mathematics to logic. Logic can be accepted in an empirical framework because logic results in analytic truths: results which are true in all possible worlds. Another way of stating this is that analytic truths are tautological, they are true by definition e.g. there is no possible world where 1=1. If all of mathematics is reducible to Peano’s axioms, themselves reducible to logic, then the logicist has successfully given an empiricist account of mathematical truth. Mathematical truth becomes tautological, that is true by definition. 1=1 is an example of a tautology, there is no possible way this relationship cannot be true, 1 will always equal 1. You do not need experience of the world to establish those truths, yet one does not need to resort to positing pure reason or innate ideas for the source of mathematics truth.

This account of mathematics also provides some epistemic comfort in the fact that Peano’s axioms are not simply left to be taken on faith. Rather, Peano’s axioms can be shown to rest on ever more base assumptions. Russell felt that this reductionist approach increased our understanding of how mathematics was structured and helped us at the other end to build more complex theories (Soames, 160). The reduction of mathematics to logic does however come at the cost of trivialising mathematics. Russell said “I thought of mathematics with reverence, and suffered when Wittgenstein led me to regard it as nothing but tautologies” (quoted in Schwartz, 16). A tautology tells you nothing you did not know already – however this may be a small price to pay to keep the rationalists at bay.

So what is the philosophical point of the logicist program? It provides an account of the a priori truths of mathematics. Firstly, it justifies the axioms of mathematics, in that it expresses the axioms of maths in terms of logical axioms which are assumed to be epistemically more certain than the axioms of mathematics. This also ensures that the rest of mathematics, provided the correct reasoning from Peano’s axioms is followed, is epistemically justified. The more partisan interest of logicism is to refute rationalist accounts of mathematics, in favour of their tautological, empiricist friendly version.

**Final Thoughts**

Returning to the question, what does it mean to reduce mathematics to logic? It means one can express the axioms of mathematics in the vocabulary of logical axioms. The philosophical point of this project was primarily to give an empirical account of mathematical a priori knowledge. This reduction also provided a justification for the axioms of mathematics, thereby strengthening the epistemic integrity of mathematics.

In the end the logicist program failed. Russell soon discovered a paradox in his theory of sets (the logic he used to express his concept of number). In his attempts to fix it, he had to add the axiom of infinity which could not properly belong to logic (Schwartz, 19). The bridge between logic and mathematics was dashed almost as soon as it was built. I think Russell hoped that if he boiled mathematics long enough he would find a rich logic base. However, as Godel’s incompleteness theory demonstrated, there is a particular flavour in mathematics that alludes reduction.

**Bibliography**

Ongley, John, and Rosalind Carey. 2013. Russell: a guide for the perplexed. London: Continuum.

Russell, Bertrand. 1919. Introduction to Mathematical Philosophy. London: Geroge Allen & Unwin.

Schwartz, Steve. 2012. A brief history of analytic philosophy from Russell to Rawls. Chichester: Wiley-Blackwell. http://site.ebrary.com/id/10560547.

Soames, Scott. 2003. Philosophical analysis in the twentieth century. Volume I, Volume I. Princeton, NJ: Princeton University Press. http://public.eblib.com/choice/publicfullrecord.aspx?p=445550.

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