Week 9a- Monday’s Comments on Arithmetic

Welcome back to the internets and civilization in general.  Please comment on the readings dealing with arithmetic and algorism.  Some questions to get you going might be….

What is number?  Is one a number?  Is pi a number?  Do you remember learning about numbers for the first time?  Plato or Aristotle or both or neither?  Are there similarities between the primary sources?  Differences of note?  Anything seem odd?

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7 thoughts on “Week 9a- Monday’s Comments on Arithmetic

  1. When reading what Boethius (via Michael Masi), as well as Al-Khowarizmi, have to say about numbers, I was struck by how they view the number one, and the similarities in opinion they show. One is not a number, but a unity. How can one be a unity? What is it a unity of? I understand the premise of one being the basic unit and there being no numbers without it, but what is one composed of? Boethius discusses numbers being created by a divine power and how the four elements and things like the changing of the stars were derived from number. Number is described as being the “principle exemplar” in the mind of the creator. Is number therefore a concept or something physical the creator thought of and brought the concept of to earth? I was interested in Boethius’ points about certain mathematical properties existing only if the other does. He uses the examples of geometry and arithmetic to show that numbers will exist without geometry, but that for geometry to exist, numbers have to. Number therefore seems like an essential idea, without which the world would not function. Is this true? Do we have to understand the concept of number for different concepts to exist, like geometry, or astronomical science?

  2. I don’t remember learning numbers for the first time, and I don’t think I’ve given numbers TOO much philosophical thought, save for the time when a teacher asked me how 1+1 could be 2 when two rain drops combined are still one raindrop. I was reminded of that instance when reading Boethius and Al-Khowarizmi specifically. It was a bit distracting that Boethius kept referring to the creator while discussing an already abstract topic like arithmetic, but his examples clarified his point that numbers are prior by nature. Numbers remain (abstractly) even when you take away geometric forms, and they also establish all things, including the ability to mapping of the course of stars. Boethius approaches arithmetic more philosophically than Al-Khowarizmi (if that even makes sense), and seems to offer two senses of a unit. Does units make up the magnitude or the multitude? He differentiates the parts that make up the two, so which would he call a unit? As for Al-Khowrizmi, as I read that everything involves numbers and that units make up everything so that unity is implied in everything, I immediately wondered if by unit he meant one, the basic quantity.

    Sacrobosco reminded me of Boethius only in that he used the phrase multitude of units in describing numbers formally. He seemed to be in line with Al-Khowarizmi in suggesting that a unit is anything said to be one. I’m still a bit confused about what he meant by describing numbers materially. What is the difference between a collection if units and a multitude of units? Is it like how Boethius distinguished magnitude from multitude?

  3. Throughout the readings I was reminiscing on my elementary mathematic education. It was a nice refresher to think about what addition and subtraction look like when you write it down to work it out. I kind of understood what Boethius was saying when he was referring to god when talking about math and arithmetic in particular. It’s really remarkable to think about how someone could understand or create a visual representation of a mathematical system without someone teaching it to him. In some ways, it does almost seem god given. Basic arithmetic is something we take for granted, learning it as children, but to create a system that works out of nothing is incredible. I would have never thought of a placeholder in a numerical system had I not been taught what a placeholder is. I also would be interested to see how mathematics with Indian numerals without a signifier for 0 would work. I’m sure it’s quite complex. It’s like the missing link of numbers and math.

  4. I found it strange that when describing the differences in the Indian, Greek, and Arabic number systems that Burnett never mentioned how it came to be that all of these early number systems were based off of ten. Who decided this was a good idea? I am not saying it is not, as I believe our numbering system today works well, but that is not to say that there could not have been another system or another option that would not work better. While it makes sense to organize numbers into organized sort of “chunks,” it is left unclear why we chose ten. Did early mathematicians simply look down at their hands and decide to base the system off of our ten fingers? And, if this is the case, why did Burnett not think it was necessary to mention it?

    While I do not remember first learning about numbers and adding and subtracting them, I can say that I am glad I had a teacher and not the Sacrobosco reading to do so. While I appreciate the detail he included and found the reading interesting, I am confused as to why Sacrobosco did not include any drawings or depictions of the operations he was describing in the essay. It seems to make logical sense to include some sort of easily understood depiction when describing a concept so abstract. While (not to toot my own horn or anything) I do fancy myself competent in both addition and subtraction, I found the reading hard to follow.

  5. Although I do not remember learning about numbers my first time I do have a distinct memory in my head of sitting in my second grade class learning about the number line and Goldie the fishes mouth () what is greater than or less than. I remember completely understanding the rules of the number line and how -7 is less than -5 but i remember not understanding why more of less than something was still less than. I always thought of numbers as rules but never actually thought about why they had come to be, so these readings were really interesting to look at math in a philosophical way.
    In the Boethius piece he discusses how arithmetic is the first of all things and how God created all upon that discipline. Its kind of crazy or impossible to think about the world without numbers – there would be no order to anything, so would anything even exist? He talks about ‘posterior’ things and for them to exist there was something or some events prior that have existed to make such event posterior.
    In the Al-Khowarizmi he says that numbers are nothing but composed of unity and therefore unity is implied in every number. Although I do understand this concept of unity he is getting at when we think about numbers being multiplied and divided how does infinity play into this? if a number can be divided an infinite number of times is unity infinity that all numbers are this way?

  6. I was intrigued by the apparently immediate recognition that the Indian number system was superior to that of the Greek and Latin systems. I imagine they (mathematicians, arithmeticians, philosophers, etc) recognized a deeper pureness or sterility of mathematics that was revealed when the symbols are not “dirty”; they carry no other connotations and can thus be used freely, without bias, fairly.

    My favorite moment was when Burnett explains how zero became essential when calculation moved from the abacus to the algorithm. What a cool moment! Zero is implicit when using an abacus; it is an absence or a lack. Zero had to become explicit, and I’m not really sure of all the significance that this has. I’ll have to think about it more, but I think this was the most tantalizing moment in all the readings for today.

    I also don’t remember learning numerals. Their forms were already deeply embedded in my world context when I was born. If someone came to me today and said “all right, these guys over here have a better way of doing this, so you have to learn these new numerals” I would probably dissolve into a panic. It would take months if not years to internalize the new symbols such that my concepts of numbers would be automatically activated by the new symbols the way they are activated by the old symbols. Hell, we can’t even adopt SI!

  7. A unit can be defined as an individual, an entire, or whole, “something.” For example, the number two is made up of two individual units. However, it seems that the definition Euclid presents an issue. Units are indivisible, in terms of Aristotle’s definition.
    But this is not what most intrigued me. In this definition of a unit, when Aristotle is being quoted, the idea of a point is brought to our attention. A unit is distinguishable from a point because a point has a relative position, something that a unit does not have: a unit is “a point without position.” I know we assign units to points in distinguishing them (such as on a graph), but as it is on its own, a unit is an analogy. This interests me because I am fascinated with dimensions and how they relate to points and figurative relationships. When I was a child, I found great flaws in the descriptions of the first, second, and third dimensions. A point, which is apparently the zero dimension; but a point inside a cube is apparently defined as the third dimension because you need three dimensions to describe it. Given that definition, it seems there would be no zero, first, or second dimension because everything in the reality in which we exist has a three dimensional relationship with the universe. If we start with a point that has no location, at dimension zero, this is reason to assume this spot does not “exist.” Something that does not have location, as described by Aristotle, is a unit, which is abstract. Then, moving to the first dimension, I have a hard time conceiving of a line being made out of hundreds of units that do not necessarily exist. They can be located in relation to each other, but then a single “point” in zero dimensions would be locatable to itself. Then from a line to a shape, the second dimension is a squared version of units… etc. I guess what I am getting at is that a unit is NOT by my understanding, distinguishable from a point if it is defined as being the zero dimension. The point is a unit, a completely figurative idea that exists as much as the number six. Six what? There is a huge disconnect between the idea of a point in an orb and a point on its own, though they are, in fact, identical, to me.
    I remember learning about numbers for the first time. I think I associated them as symbols very early in life, maybe by age two or three. I’ve always been fascinated with, and very good at, math. I remember in first grade going through numbers in my head, adding them to themselves until I couldn’t go any higher: 1 + 1 = 2, 2 + 2 = 4, 4 + 4 = 8, 8 + 8 = 16, 16 + 16 = 32, etc. into the thousands. I also remember being obsessed with graph paper and counting boxes as I colored them in or filled them with X’s. I was always better with abstract math, such as algebra or calculus, whereas geometry proved to be more difficult. Perhaps this is why I have such a hard time with dimensions and points, especially when applying it to the concept of the abstract “unit.”

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