Embodied Mathematics


Read here the embodied explanation of mathematics, and in particular, exponentiation. The embodied comprehension of the world is a critical part of comprehending scale, as we relate all things to our own bodies and realm of experience (see also Narcissism).

The embodied explanation of exponentiation is based on the chapter “what is e” in Where Mathematics Come From by Lakoff and Nunez.
Other physical, biological issues are presented.

4Gs of embodied arithmetic


The concept of order of magnitude is traced to the 4G’s of embodied arithmetic.

Based upon our interaction and experience with the real world, we have four basic metaphors with which we conceptualize the natural numbers (and partly also fractions):
  • Object Collection
  • Object Construction
  • Measuring Stick
  • Motion Along a Path

Lakoff and Nunez, in their book "Where Mathematics Comes From" term these the “4Gs” due to their attributed importance to the whole of mathematical thinking (the “G” stands for grounding.) It is perhaps worthy to point out at this point that these metaphors do not represent truths per se about the world, but rather they try to express how we humans think about certain ideas and in this case, mathematics.
The last is added to complete another metaphor, namely NUMBERS ARE THINGS IN THE WORLD.

Closure


In addition to the 4Gs, the community of people practicing arithmetic also perform "closure" on groups of operations, seeking to tame monsters by defining operations that at first do not seem to be defined. For example, three to the zero power is defined as one: that is an operation of closure on the function of exponentiation.

Embodied Exponentiation


Limitations Of The Conventional Explanation

Exponents are conventionally understood as multiplying a number by itself a set number of times.
The expression 2^5, two raised to the fifth power, is usually taken to mean 2 * 2 * 2 * 2 * 2—two multiplied by itself repeatedly. But the corresponding expression e^pi, the number e = 2.718281828459045 raised to the power pi, where pi is 3.1415926535 and I is the second power root of -1 cannot mean e multiplied by itself pi times and the result multiplied by itself I times.
Raising to an exponent cannot, then, be understood in all cases as self-multiplication.

Logarithm Base 10

A logarithm is a mapping that enables calculation of multiplied numbers by adding. Invented by John Napier in a time where mechanical calculation was not available.
Napier began from the pattern that q*q*q*q*q = (q*q) * (q*q*q) = q^2 * q^3 = q^(2+3) = q^5. A table can then be generated,
Number
Logarithm
1
0
10
1
100
2
1000
3
10000
4

The table specifies a mapping that has been named logarithm or log for short. The table maps real numbers onto real numbers and products onto sums. That is, it maps a product a*b onto a sum a’*b’ by mapping number a and b onto a’ and b’ and by mapping the operation of multiplication onto the operation of addition.

Negative Logarithms From Closure

Logs map the operation of multiplication onto the operation of addition. To transducer the operation properly, identities and inverses need to be encoded properly.
Identity: the multiplicative identity 1 is mapped onto the additive identity 0.
Inverse: the multiplicative inverse 1/a is mapped onto the additive inverse –a’.
Given this closure we can now extend the table:
Number
Logarithm
1/10,000
-4
1/1,000
-3
1/100
-2
1/10
-1
1
0
10
1
100
2
1,000
3
10,000
4

Fractional Logarithms From Closure

As an abstract mathematical function, log maps every positive real number onto a corresponding real number, and maps every product of positive real numbers onto a sum of real numbers. However, this mapping does not a priori provide an algorithm for computing such mappings for all the real numbers. Approximations to values for real numbers can be made to any degree of accuracy required by doing arithmetic operations on rational numbers.
A full summary of the constraints of this mapping are:
*1 is mapped onto 0
*10 is mapped onto 1
*10^n is mapped onto n, for all real numbers n
*Products are mapped onto sums

Logarithmic mapping for base_10 by Lakoff
embodied_math_Logarithmic_mapping_for_base_10_by_Lakoff.png


Logarithm Base B

While historically, base 10 was the first exponential system, as embodied by Hindu Arabic place notation, in fact 10 is an arbitrary choice. The multiplication – addition mapping works with any number used as the starting point, called the base.
The chess story is a famous example that uses the number 2 as the multiplier. It would generate the following table:

Number
Logarithm
2^n
N
1/4
-2
1/2
-1
1
0
2
1
4
2
8
3
16
4
32
5
64
6
128
7
256
8

For logarithms with base 2, the constraints above are rephrased as:
  • 1 is mapped onto 0
  • 2 is mapped onto 1
  • 2^n is mapped onto n, for all real numbers n
  • Products are mapped onto sums
And the general form, for any b greater than 1, would read,
  • 1 is mapped onto 0
  • b is mapped onto 1
  • b^n is mapped onto n, for all real numbers n
  • Products are mapped onto sums

embodied_math_Logarithmic_mapping_for_arbitrary_base_b_by_Lakoff.png
Logarithmic mapping for arbitrary base b by Lakoff

Exponentials

The mapping of multiplication to addition, being one-to-one, is valid in the reverse direction. The inverse mapping will map a real number n onto a positive real number b^n. The constraints of this exponential mapping are:
Table below compares logarithmic and exponential constraints.

Logarithmic mapping from multiplication to addition
Exponential mapping from addition to multiplication
1 is mapped onto 0
0 is mapped onto 1
b is mapped onto 1
1 is mapped onto b, for b>1
b^n is mapped onto n, for all real numbers n
n is mapped onto b^n, for all real numbers n
Products are mapped onto sums
Sums are mapped onto products

For example, given b=2, we have the inverse of the table above:
Number and logarithm mapping base 2
Exponent
Value
n
2^n
-2
1/4
-1
1/2
0
1
1
2
2
4
3
8
4
16
5
32
6
64
7
128
8
256
For this version of the table, the values can be computed by self-multiplication. But the exponential case applies to an endless number of cases where this is not true. Consider, for example, 2^1/3. This is a well-defined instance of the exponential mapping with base 2—namely, 3rd root of 4—but it is not an instance of 2 multiplied by itself 2/3 of a time, whatever that could mean. Similarly, 2^pi is also a well-defined instance of the mapping, but it, too, is not an instance of 2 multiplied by itself 3.1415926… times.
Now the difference between self-multiplication and exponentiation is made clear. When we write 32 = 2*2*2*2*2 we are using the 4G Arithmetic is Object Construction understanding 2^5 as constructed by multiplication. The symbolization ^5 is ambiguous. It can mean either self-multiplication (construction) or exponentiation (mapping). Exp2(5) would be a clearer way to show that this is a mapping function. In this notation Exp2(pi) makes the same sense, being “pi is the input to the exponential mapping and 2 indicates what 1 maps onto in this version of exponential mapping.”

Natural Logarithm E

To review: an exponential function is a mapping of the operation of multiplication onto the operation of addition. The notation y=b^x is read as “y=power(base b)^x” or “ the result of the mapping of addition onto multiplication where 0 is mapped to 1, 1 is mapped to b, and b^x is mapped to y.
Now let’s consider these values in pairs. That is, two values on the addition scale, and the values they are mapped to on the multiplicative scale. Each value changes by a value delta. If the first value is y=b^x and the second value is y’=b^x’, then we are comparing the change in the y values, which is b^x’-b^x, to the change in x, which is x’-x.
Here is an example in base 10: Rate of change for base 10, from 1 to 2
First Value, y=b^x
Second Value y’=b^x’
Change in y
y’-y
Change in x x’-x
The ratio of the change, (y’-y)/(x’/x)
10=10^1
100=10^2
90
1
90
Let’s zoom in on the rate of change. Instead of jumping from 1 to 2, let’s go from 1 to 1 1/2, and even smaller intervals. Notice that the rate of change converges on the value 23.
Table 6 Rate of change for base 10, for ever smaller increments in x
First Value, y=b^x
Second Value y’=b^x’
Change in y
y’-y
Change in x
x’-x
The ratio of the change, (y’-y)/(x’/x)
10=10^1
100=10^(2)
90
1
90
10=10^1
31.62=10^(1.5)
21.62
0.5
43.25
10=10^1
17.78=10^(1.25)
13.84
0.25
55.36
10=10^1
12.59=10^(1.1)
3.24
0.1
32.60
10=10^1
10.23=10^(1.01)
0.24
0.01
23.83
10=10^1
10.02=10^(1.001)
0.02
0.001
23.11
10=10^1
10.0023=10^(1.0001)
0.0023
0.0001
23.034
10=10^1
10.0002=10^(1.00001)
0.00023
0.00001
23.0266
As the change becomes smaller, the ratio of change converges on the value 23, a number that is not the same as the base 10.
Is there a number for which the ratio of change converges on the value of the base?
There is, this number has been named e. E is defined as 2.718281828…
Here is the same chart with e as the base. The numbers are more difficult to read, but the idea is the same:
NEED TO RECALCULATE THIS TABLE
First Value, y=b^x
Second Value y’=b^x’
Change in y y’-y
Change in x x’-x
The ratio of the change, (y’-y)/(x’/x)
e=10^1
7.389056=e^(2)
12.696
1
12.69648082
e=10^1
4.481689=e^(1.5)
2.907
0.5
5.81473406
e=10^1
3.490343=e^(1.25)
0.991
0.25
3.96538445
e=10^1
3.004166=e^(1.1)
0.3160
0.1
3.15950899
e=10^1
2.745601=e^(1.01)
0.02759
0.01
2.75937489
e=10^1
2.721001=e^(1.001)
0.002722
0.001
2.72236242
e=10^1
2.718554=e^(1.0001)
0.0002717
0.0001
2.71868960
e=10^1
2.718309=e^(1.00001)
0.0000271832
0.00001
2.71832260
As the change becomes smaller, the ratio of change converges on the value 2.718281828, a number that is exactly the same as the base e, which is defined as 2.718281828…
Think about that: e is the unusual number, and the only number, that when mapping addition onto multiplication, if you zoom in to the smallest possible change, then the rate of change at that tiny increment is equal to the value you get when you map “1”.
This is an interesting claim when you consider mapping addition to multiplication is the same thing as a growth condition.

Lakatos version of embodied exponentiation


It is a natural tendency for modellers to justify and preserve their models, even when confronted with apparent exceptions. Meanwhile, other people question and challenge the model, for three reasons.
  • to better understand the model
  • to increase confidence in the model
  • to improve the model
Imre Lakatos (1922-1974) in his book Proofs and Refutations proposed this pattern, stressing that the exception "proves" the rule. This statement has several meanings – some absurd and some quite useful. Among other things, it means that a robust model is one that stands up to repeated attempts to disprove by exception.

Pease and Smaill offer the following version in Lakatosian terms.
  • Conjecture 1: The Arithmetic is Object Collection metaphor matches our characterisation of number.
  • Counterexample: In the number domain we have a concept of 7-7, but there is no corresponding collection.
  • Concept-stretching solution (the opposite to monster-barring): widen the concept of collection to include the empty collection.
  • Conjecture 2: The measuring stick metaphor matches our characterisation of number.
  • Counterexample: The measuring stick metaphor allows us to form physical segments of a length which doesn’t correspond to any number.
  • Concept-stretching solution: widen the concept of number to include irrationals.
  • Conjecture 3: the arithmetic as motion along a path metaphor matches our characterisation of number.
  • Counterexample: the points found when moving backwards from the origin do not correspond to any number.
  • Concept-stretching solution: widen the concept of number to include negatives.

Alternatively, we might have a conjecture like:
  • Conjecture 4: numbers always correspond to something in the physical world.
  • Counterexample: irrational numbers do not correspond to anything in the physicalworld (assuming the Arithmetic is Object Collection metaphor).
  • Monster-adjusting solution: irrational numbers do correspond to something in the physical world (if we assume the measuring stick metaphor).

It is not clear to me whether this is a useful way of seeing the metaphor refinement, or the refinement can ever be described in methods other than concept-stretching or monster-adjusting.

Biological Basis Of Exponentiation


Dehaene advances a new paradigm, that logarithmic evaluation is more natural or intuitive than linear. This accords with the networked nature of neurons and organic growth, each of which form the organic infrastructure of thought and perception. See for example Fechner's law.