Sunday, November 6, 2011

Finite and Infinite

I have stated many times that conventional mathematical appreciation is based on a merely reduced notion of the infinite (where effectively it is treated as an extension of the finite).

This for example defines the nature of conventional proof where what is true for the general case (potentially applying to the infinite) is thereby assumed to apply to all actual (finite) cases.

And I have referred to this basic reductionism as interpretation that is linear (i.e. 1-dimensional) in qualitative terms.

This linear approach is also very much to the fore in the treatment of series where in most cases once again a seemingly unambiguous relationship as between finite and infinite emerges.

For example from a finite perspective we can see that a geometrical series such as 1 + 1/2 + 1/4 + ... converges towards some finite limit (getting ever closer to 2 without actually reaching this limiting value).

Therefore when we say that in the limit the value of the series = 2 (where the no. of terms is infinite) this again seems to comply with linear type interpretation (i.e. where the infinite is treated as an extension of finite notions).

However there are other cases where what appears true in finite terms does not readily comply with infinite notions.

For example

1/(1 - x) = 1 + x + x^2 + x^3 + x^4 + ..... (where the number of terms in linear terms is assumed infinite).

This would imply therefore that when x = - 2 that

1/3 = 1 - 2 + 4 - 8 + .....

However when we view the R.H.S in finite terms, we can see that the terms get progressively larger with the value of the series diverging. Now it is true that the terms (and consequent sum of terms) alternate between positive and negative values. However from a linear perspective we cannot say that its sum will converge to a definite finite value.

However the - apparent - equivalent L.H.S expression suggests that this is precisely what happens.

So once again from a linear perspective we obtain a result that is intuitively not in keeping with its rational mode of interpretation.

This strongly suggests that a different form of rational interpretation is required to explain the nature of the result.

However because it is qualitatively defined in terms of 1-dimensional interpretation, Conventional (Type 1) Mathematics is not appropriate for this task.

Now when we return to our example we can see what is the problem

If we consider 1 + x + x^2 + x^3 + x^4 + .... + x^(n - 1), where the series is defined in terms of a finite number (n) of terms,

then 1 - {x ^(n - 1)}/(1 - x) = 1 + x + x^2 + x^3 + x^4 + .... + x^(n - 1)

So the conclusion that

1/(1 - x) = 1 + x + x^2 + x^3 + x^4 + ..... (where n is infinite), is based on the the assumption that n - 1 = n terms (when n is infinite).

Thus the logic that applies for the infinite case i.e. n - 1 = n is directly confused with standard finite logic i.e. n - 1 ≠ n.

This then leads to the non-intuitive result (in linear rational terms) that for example

1 - 2 + 4 - 8 + .... = 1/3

Though mathematicians are of course aware of this anomaly, they attempt to explain it in terms of two results that comply with differing domains of definition.

However this avoids the deeper qualitative question of what such non-intuitive results actually entail!

Facing up to this issue requires accepting the radical conclusion that just as numbers such as 1, 2, 3, 4, ... etc. have a well-defined meaning in quantitative terms, equally they have an - as yet - unrecognised meaning in qualitative terms whereby they refer to unique modes of rational interpretation of symbols.

Once again Conventional (Type 1) Mathematics attempts to confine interpretation to 1-dimensional logic in qualitative terms. However potentially an unlimited set of other qualitative logical interpretations can be given.

And the series that I have used to illustrate this point itself points to the need for a different means of rational interpretation (so that results can then intuitively concur with the correct rational mode adopted).

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