Martin Bryan, IS-Thought
Mathematicians tell is that 2 + 2 = 4 is a mathematical rule, yet this rule is not inviolate. For example, it is also mathematically correct to say that 2 + 2 = 11, or 2 + 2 = 10. The only difference between these three statements is the base used for the mathematical analysis (10, 3 or 4). But other ways of expressing the rule could be II + II = IV, ௨ + ௨ = ௪ (using Tamil numerals) or even II + II = IIII. But how does this differ from II + II = :: or : + : = □? The symbols we use to represent the statement do not affect it. We could even use the Arabic right-to-left notation ٢ + ٢ = ٤. Now when I read any of these statements I think "two and two is four". This matches the way I record the meaning of the symbols in my mind. But a French mathematician would think in terms of "deux plus deux est quatre". His internal mapping of the symbol to a thought within his mind differs totally from mine. How do we equate all of these options? Would it be equally valid for a multilinguist to think in terms of "deux i duo is fier"? Arabic numerals and mathematical symbols, together with an unspecified agreement between mathematicians to use a base of 10 that is inherent in the use of Arabic numerals, provides a representation of mathematical statements that is independent of the language of the reader, but does not indicate in any way the way in which we internalize the process of addition. I do not need to use words to describe the concept. If I hold up two fingers on one hand and two fingers on another hand, and then hold up 4 fingers on one of the hands and none on the other I can also indicate the concept of addition without using words in a way that another human, or any intelligent animal, can understand.
Are there any limits to the way in which the process of addition can be applied? There are only two that I can identify. Theoretically there is no limit to the number of numbers you can add together. You can add 2+2+2... as many times as you want. We have a shorthand for this, 2n, which allows us to describe any number of additions of 2. But what if the number of additions is infinite. Is the answer 2∞? If so what does twice infinity mean? Or is it 2(∞−2)? If do what is two less than infinity? The other limit concerns what it is we are adding. If I have two apples and two pears I have four fruits, but only 2 apples and 2 pears. In mathematics we use concepts such as 2A + 2B = 2B + 2A ≠ 4A or 4B, but could equal 4F.
Mathematics is built on the concept that we are working with a constant "it", the concept of 1. But what is "one"? One is simply a unit of measurement along an axis of measurement. As long as we restrict ourselves to this measurement axis we can apply the mathematical concept of addition. It is simply a movement in the direction of increase along the axis from a starting point defined by the first number in the statement by the number of increments given in the second part of the statement, resulting in an end-point as specified in the result part of the expression.
But what if we want to move in the other direction? That's easy, we use the concept of subtraction. But how can we represent this. When we teach children subtraction we start by talking to them in terms of taking away. We say "four take away two leaves two". We demonstrate this by showing them 4 matching objects, which we count, and then removing two to them before counting the result. But is this an ambiguous demonstration of subtraction, or is it just as valid as a demonstration of halving? If I take remove one object from a pair, I have the same problem. If I take away 1 object from a set of 4, and then 2 away from the resulting set of three then I have developed a less ambiguous definition of subtraction (though its still not perfect). But what happens when I then want to take two away from 1 to illustrate the concept of −1? Once we remove the one object from our demonstration we are left with an empty space. How can we remove anything from an empty space? Is there a practical representation of −1 other than a measurement along an axis away from the end of an axis of positive numbers?
Does the way in which we represent subtraction in terms of words help in any way. What if we change 4−1=3 to IV−I=III, or ௪−௧=௩? Does it not matter what we use to represent the terms? If we replace "three" the "number 3" and "one" by "meal" and "four" by "Chinese" what happens to the phrase "four take away one is three"! Is this still a valid mathematical statement? What about "quatre moins uno is tre"? I contend that both of these examples show that the way in which we represent the concepts is significant.
Before going on to consider the problems related to other operators, let us just return to the representation of the empty set, zero or 0, the key to the success of Arabic numerals. Unfortunately this number is the one that breaks every rule. For addition and subtraction the concept of "a movement in the direction of increase (or decrease) along the axis from a starting point defined by the first number in the statement" cannot apply. When zero is used as the second of the operators no movement takes place? When it is the first number there is movement, starting from the axis origin. But what about 2−2 = 0? Here what is happening is a movement back to the origin, or the removal of objects to create an empty set/space. When it comes to multiplication we have an even bigger problem, as whatever we multiply by zero we end up at the origin of our counting axis, while if we try to divide anything by zero we, if our computer does not bulk at the instruction, end up with the mythical ∞ sign.
So what about other multiplications and divisions. Are they really logical? Does the way mathematicians express fractions matter? For example, what is the difference between 3 ÷ 4, 3×4-1, ¾, 0.75 or "threequarters"? Is there a conceptual difference between a "wing threequarters" and a mathematical one? What is 2/∞? Is 1/π really a rational number, or is pi just too irrational to count? What is 180/π and how does it differ from 1 radian? As far as I can see both radians and the pi sign are simply mathematical conveniences dreamt up to avoid the irrationality of processes based on circles when measured using rational numbers. Neither have been, or can ever be, assigned a definitive value. In fact very few fractions can be expressed rationally. Try expressing 73/47 or 47/73 as decimals, or 0.151515 as a fraction to convince yourself that there is no real relationship between decimal numbers and fractions caused by division. Ask yourself what the difference is between -73/47 and 73/-47 and how you can represent either of these in real life. Does -π have any physical representation?
For multiplication mathematicians have introduced different conventions depending on whether you are multiplying the same number (2n), decreasing consecutive numbers (5!) or random numbers (2×5×7...). The first two of these conventions are limited. For example, is 2∞ valid or should 2(∞/2) be the largest permitted power? What is the maximum number permitted for n! which does not provide a result that exceeds ∞? Is there any limit to the number of numbers that can be multiplied together in the lifetime of the universe? Is 1.2.3.4.5.6.7.8.9 the same as 9! or simply a list of numbers? Can all forms of multiplication be represented using roman or Japanese numerals? Once again, we come across the fact that mathematical conventions have unstated limitations which we are all too prone to ignore.
The claims made for logic are just as spurious. Using logic 10 AND 10 = 10, but 4 AND 4 ≠ 4. The ambiguity in the AND operand can also be seen by comparing "Marks and Spencer" to "Marx and Spenser". The everyday word "and" is one of those Humpty Dumpty words that means what we want it to mean, no more and no less. 2 AND 2 may equal 4 to a mathematician or a child, but is meaningless to a logician. But a child could equally well say "two and two is twenty-two". But if we were using : to represent two units would :: represent 4, 22 22, or is it simply a square pattern?
Logical OR, or Exclusive OR (XOR) is just as illogical! 11 OR 11 = 11 but 11 XOR 11 = 00. But if I say "four or five" the answers is not a number but a range, and I cannot give a clear result to the statement.
What is the point of this diatribe? I hope by now you have begun to question some of the assumptions made by mathematicians. Yes, maths can be a useful tool for recording the way in which events interact, and for predicting how they will interact in the future, but unless you clearly understand its limitations you can get into very sloppy ways of thinking that make unreasonable assumptions about the world around you. Maths is a tool for modelling what might happen in the real world, but unless we adopt a correct model to fully predict events we cannot expect maths to solve problems for us.
November 2002