(Editor's note: not all symbols used here may appear as normal, so may appear spelled out.)
Notice that every numerical concept can be symbolized. And symbolizing a numerical concept allows the use of that concept within a mathematical expression. For a simple example there is the concept of two, which is normally symbolized as 2. For more complex examples there are the concepts of constants and variables: We can say that n is some constant number, where we have it be the same number every time n is used in following expressions; and we can say that x is some variable number, that can be any number at all so long as its the same in the same expression. With all these symbols we can make a complex expression such as:
2xn + 3x - n = 8.
There are a whole lot of different numbers that will fill in for n and x and make this expression true, so that everything on the left of the equality will be equal to the 8 on the right. But any way we fill in for x and n, we have there a mathematical expression utilizing some numerical concepts. So we have started with numerical concepts and mathematized them.
There are numerical concepts that mean something very specific, and accordingly, are specially symbolized. For example: The ratio between the circumference of a circle and its diameter is symbolized as pi. So we can take that concept, that ratio, symbolize it, and fit it into an expression such as the following:
3pi > 2pi + 1.
This expression is of course true, due to the numerical value of the ratio of the circumference of a circle to its diameter (namely that value is approximately 3.14159 . .)
These are all examples of a natural progression from symbolizing a numerical concept to mathematising it, fitting it into expressions. However, there is a problem with this natural progression in that it breaks down when it comes to some specific numerical concepts.
(1) Consider the numerical concept of infinity (for example, as the length of the series of real numbers). That concept is symbolized as ∞. Here we have a numerical concept and then a symbol for it, and so we think we can use it in mathematical expressions. An example of such an expression would be:
3∞ > 2∞ .
On the surface, this seems to be an OK thing to say. But beneath that surface there appear some problems with it. Note that all infinite series can theoretically be counted, and counting means making a one-to-one pairing with the natural numbers (that's 1, 2, 3, and so on.) This pairing is theoretically possible, but of course it is hard to do completely, since it would take all time to do it, an infinity of time to be exact. Now there is something unique about one-to-one pairings: If two sets can be paired one-to-one with each other, it must be that there are of the same length. So this means that any infinite series must have the same length as the series of natural numbers.
Now suppose a series A (a series of something, no matter what it is) can be paired one-to-one with the series of natural numbers, and another different series B can also be paired one-to-one with the series of natural numbers. It must be that A and B are the same length, even if they have different contents. This should be so since if two things each have the same length as a third thing, they have the same length as each other. Since all infinite series can be treated like this, paired one-to-one with the natural numbers, so they all are the same length. What makes us think they have different lengths is that they have different contents, but the fact that their contents differ says nothing about their length.
But then re-consider the above expression that was used as an example with this numerical concept of infinity:
3∞ > 2∞.
Note that this expression must be false, because 3∞ and 2∞ are actually equal, as every infinite series is of equal length. Here then is a breakdown in the mathematising of numerical concepts; a case where a numerical concept can certainly be symbolized, but yet it cannot be utilized within mathematical expressions. We can create mathematical expressions with this concept, but those expressions end up being bad, bad in the sense of being false.
(2) For a second example of problems in mathematising numerical concepts: Consider the concept of imaginary numbers, symbolized as some number multiplied by i. This i is the square root of -1 (or in shorthand √-1); that is to say, the number which when multiplied by itself will result in the number -1. Thus it is said that the square root of -36 is 6i. (Or in shorthand √ -36 = 6i.) This is accomplished by saying that -36 is 36 multiplied by -1 and thus that √ -36 is √36 x √-1, which is then shortened to 6i.
We have a clear case here again of a numerical concept, the square root of -1, and a symbol for it, i, and then sooner or later have the opportunity for using that symbol in expressions, such as this:
6i > 3i.
However, the big problem with this or any other expression involving the imaginary numbers is that there is no such thing as i; there is no number at all which when multiplied by itself results in any negative number, let alone just the number -1. Another way of saying this is that the set of numbers which when multiplied by themselves result in a negative number is the empty set. Any negative number multiplied by itself results in a positive one, just like positive numbers multiplied by themselves do. Another way of saying this is to say that this is a case of a numerical concept with no numerical value.
So re-consider this expression again:
6i > 3i.
Since there is no number that is i, 6 multiplied by it would have no numerical relation to 3 multiplied by it, and so this statement must be false. But notice that we can say the same about these following expressions:
6i < 3i ;
6i = 3i ;
we can also say that these are false too.
Now someone may try to counter this line of thinking by saying something like this: You know what we would get if we add 2 unicorns and 3 unicorns? We get 5 unicorns, and we get this even if there are no such things as unicorns. Now of course it is true that 2 unicorns + 3 unicorns = 5 unicorns, as it is true that 2a + 3a = 5a for any a. But these statements are true because they work out as statements about objects and numbers. There are no unicorns here in our world, but the unicorn is a possible thing to have show up some day, or to find somewhere else in another place different from ours, so it is at least a possible object. And its true that 2a + 3a = 5a for any a, even where the value of a is zero. But this works out because zero is actually a number just like other numbers. But i is not an actual number, since, again, the set of numbers which when multiplied by themselves results in -1 is the empty set. The numerical concept here is actually neither a number nor a possible object, so would not fit in as an a in the expression 2a + 3a = 5a for any a.
Again in the end, we have an example of a numerical concept which, though it certainly can be symbolized, cannot be used in any mathematical expressions without causing trouble.
(3) For a third example of problems in mathematizing numerical concepts, consider division by zero. As you will recall, that's not division of zero, which is always zero. This would be some number, say 14, divided by 0 or 14 ÷ 0.
Suppose we say that the zilch number, symbolizing it as zn, is that number which results from division of some number n by 0. So 4 ÷ 0 is symbolized z4. Call that the zilch number of 4, and by doing that we have invented a way of symbolizing division by 0. Suppose we take the common track of thinking that by symbolizing division by 0 we can then mathematize division by 0, and thus use it in expressions. As a result we can consider the expression:
z10 > z5.
The problem with this statement, of course, is that it is false. . Clearly z10 is not only not greater than z5 but it has no mathematical relation to z5, because there are no such numbers as z10 or z5 or any zn.. Division by zero is a numerical concept but it has no numerical value. Of course note that we say the same about these:
z10 < z5 ,
z10 = z5 ,
these as well are false. Any expression containing any zilch number would be false no matter how derived. Again we have an example of a numerical concept which can be symbolized but yet cannot be used in any mathematical expression.
There is one basic conclusion that can be made about the symbolizing of numerical concepts: Given that there are at least three examples of cases where the symbolization of numerical concepts does not permit the mathematising of them, there is no telling how many there are besides three. So there has to be some filtering of which numerical concepts are allowable within a system and to what extent they are allowed. Some numerical concepts can fit into our mathematics without restriction, but some don't, maybe by deriving results that are false (as above) or even something as simple as not following all the standard rules, for example, by being additive. And we should not be surprised about this, since one of the most basic numerical concepts, that of zero, has some of the hallmarks of a problem case. We already know that, for example,
2 x 0 = 4 x 0
but also know that
2 x 0 + 2 x 0 = 4 x 0.
But if all numerical concepts are additive this should not happen. If an expression on the left of an equality is equal to an expression on the right, then adding to the left should unbalance the equality, but in this case it clearly does not. This just means that in the case of the numerical concept of zero there is a limit to the expressions or operations in which it is allowed to have mathematical consequences.
A general overall conclusion of all of this is that not all numerical concepts can be mathematized. And so it must be that any system of mathematics that incorporates every numerical concept is unsound. This is to say: Any mathematical system that admits every numerical concept will allow the derivation of theorems (results) that are false. It is certainly true that allowing in any and every numerical concept will allow the derivation of any result we may desire, and some of these results may be interesting. But the fact of them being interesting does not make them true, which means they cannot be trusted entire.
(published 6/2/19)
Reader Comment: It’s not clear that this is related or not to a different idea, that of completeness. The idea is that a system is sound if all results derived from it are true, and unsound if it is possible to derive results that are false; whereas a system is complete if all true statements that are expressible in the language of the system can be derived within it, and incomplete if there are true statements that cannot be derived.
Writer Response: That’s actually dead-on right; this is intended to be only on soundness, (and that of some systems) and related to completeness as you described. The issue of completeness is its correlate.
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