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• Lonergan's Insight
3/3/2007: The Viewpoint of Arithmetic (this expands
one part of the February 17th outline)
Dr. David P. Fleischacker
Click here for a PDF version of these notes. (as a note, there are a
few grammatical corrections below which are not found in the PDF version)
- The Meaning of “1”
- A sheep
- A single Sheep
- A unified subject of characteristics
- Same even in change
- The Residue of individuality
- uniqueness of data that are unified
- empirical residue
- basis of numerical 1
- Numerically one sheep
- Unity does not equal 1
- Saying something is the subject of is not the same as saying it is 1.
- Individuality does not equal unity
- Saying something is unique is not the same as saying it is the unity of characteristics.
- Unity of data vs uniqueness of data
- THIS unity or THAT unity
- Math starts with distinctness, uniqueness, individualness
- Individuality does not equal 1
- an individual thing does not equal one individual thing
- this sheep, this car, this house
- that sheep, that car, that house
- From individual to 1
- need to “put together" or "group" this and that
- Needs addition to mean numerical "1" or
"2", etc., etc., etc. (needs that relation of being “put together” with another)
- Fido, King, Joker, and Jezebeel
(each a dog-- each an individual unity vs. 4
dogs)
- How many? Which one is first? (another
way to think of this is to notice the
difference of the questions. The
question leading to the insight of unity or
the recognition of individuality is
different from that leading to the
mathematical number.
- As a note, two important theological areas that Lonergan
deals with individuality, unity, and number is in his
Trinitarian theology (unity and distinctness of the Father,
Son, and Holy Spirit) and in Christology (the unity of the Son
incarnate).
- Rules in Arithmetic
- Deductive expansion: expanding one’s “view” of numbers via the same operation
- Positive integers
- Addition tables
- Homogeneous Expansion: expanding the types of operations
-
Operations |
Inverse Operations |
Addition
>>>>
|
Subtraction
<<<<
|
Multiplication
>>>> |
Division
<<<< |
Powers
>>>> |
Roots
<<<< |
- All these operations are defined ultimately in terms of addition
- Subtraction: inverse of addition
- Multiplication: adding a number so many times to itself.
- Division: inverse of multiplication (finding the number which when added to itself so many times gets the
number that is being “divided”).
- Powers: multiplying a number by itself so many times
- Roots: inverse of powers (finding the number which when multiplied by itself so many times gets the number
that is being “rooted”).
- The Arithmetic Viewpoint
- The Deductively Expanded Arithmetic Viewpoint
- Notice how the initial operation of addition, which then gives
"meaning" to number allows one to create a viewpoint identified as the positive integers. Another way of saying this is that it allows for the creation of the “world” of the positive integers.
- The Homogenously Expanded Arithmetic viewpoint
- With the addition of subtraction, one is then able to move backward and forward in the world of the positive integers, and then add to that world the world of the negative numbers.
- With the addition of the operations of multiplication, division, powers, and roots, one’s viewpoint and world of positive and negative numbers not only expands in how one gets to these numbers (through these other operations), but adds to these integers the worlds of fractions (by division) and of surds
as well as imaginary numbers (in roots).
- Some of these activities bring questions such as what happens when one adds two negative numbers, subtracts two negative numbers, or divides a negative into a positive number. This is just the beginning of questions that will lead into the higher viewpoint of algebra.
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