I am aware in these blogs that I am often repeating points that I have made in other entries. However as I am attempting to open up what I believe is entirely new ground with the most fundamental consequences possible for the true nature of Mathematics, I understand that initially considerable resistance may exist to accepting the implications of what is being conveyed. So in continually approaching the same problem from a number of different angles it may therefore better enable interested readers to grasp the central argument.
Yesterday I pointed to the crucially important fact that two notions of number exist (cardinal and ordinal) that are quantitative and qualitative with respect to each other.
Secondly I stated that we must define the number system in a dynamic interactive manner to preserve the distinctive nature of both aspects.
And as the conventional mathematical approach attempts to define number in a static absolute fashion this leads to gross reductionism with respect to ordinal notions; indeed ultimately - because cardinal and ordinal are interdependent - it also leads to considerable distortion even with respect to the cardinal notion of number!
And as it is so important, let me once again return to the central point.
In actual experience when we use a number - say the number "4", an inevitable interaction with respect to two distinctive type of meaning are involved.
So from the cardinal aspect we understand 4 in a collective whole manner. This is indeed why we refer to the natural numbers as wholes i.e. integers.
However a second distinctive ordinal aspect is likewise involved. From this perspective 4 is seen - not directly in its collective indivisible state - but rather in terms of its individual members.
So 4 now is defined ordinally in terms of its 1st, 2nd, 3rd and 4th members.
Now we saw yesterday that there is a fundamental problem with the cardinal approach to this issue.
From this perspective 4 = 1 + 1 + 1 + 1.
However this attempt to express the collective whole number 4 (in cardinal terms) as the sum of individual units (that are also cardinal) robs the ordinal notion of number of any coherent meaning.
For if all the units of 4 are homogenous (in quantitative terms), then we have no means of consistently ranking these numbers as 1st, 2nd 3rd and 4th which applies some unique means of applying qualitative distinction.
For example in the Premier league the top 4 clubs qualify to play (though last year was an exception) in the European Champions League.
So when we say that 4 clubs can qualify, we are using 4 in its collective cardinal sense (that is quantitative). However when we look at the 1st, 2nd, 3rd and 4th members we have now switched to a unique ordinal sense of interpretation (that is qualitative in nature).
Thus the very means of distinguishing these members requires identifying unique qualitative features. for quite simply if all members are the same - as the cardinal approach would suggest - then no means exist for attempting any unique ranking!
So this problem with respect to distinguishing cardinal and ordinal relates to the fundamental issue of properly defining whole and part notions.
In conventional Mathematics a reduced approach is adopted, whereby the whole is viewed as merely the sum of its quantitative parts. In this way ordinal notions are directly confused with cardinal in the mistaken view that number can be interpreted in a merely quantitative fashion! So a notion that strictly applies to the independent nature of number is mistakenly applied to the interpretation of the relationship between numbers.
When one looks more closely at the true ordinal nature of number one quickly realises that it has a merely relative identity that crucially depends on context.
For example if I attempt to identify the 2nd of two members, then it may indeed appear to have an unambiguous identity. However if I now identify the 2nd of 3, the context has now changed so in fact a different relationship of the ordinal member with other members is involved.
And of course ultimately a potentially unlimited set of such definitions of 2 as ordinal exist as the 2nd member can be defined in terms of 2, 3, 4, ....n members (where n has no finite limit).
We are now accustomed to use numbers (to refer to ordinal rankings) in a wide variety of contexts. Thus in sport for example we have the PGA rankings in golf. We have similar rankings in tennis, football, boxing - indeed whatever sport you might consider. Then we have economic rankings of shares, GDP per capita and so on.
For example we might say that Tiger Woods is ranked no. 2 in men's golf at present.
However this use of 2 here refers to an ordinal rather than cardinal meaning, the interpretation of which changes (depending on context). If there were only two golfers ranked in the world it might appear that we can give 2 (in this ordinal context) an unambiguous interpretation. However with world golf now so competitive with a great many professionals seeking a ranking, 2 in this context thereby conveys a very different meaning.
Therefore the key point being made here is that the meaning of 2 - when used in an ordinal sense - is merely of a relative nature depending on context. It literally depends therefore on its relationship with other numbers.
And because cardinal and ordinal interpretations are inevitably linked in a dynamic interactive manner, this implies that the cardinal notion of number is likewise of a merely relative nature. Thus though we focus directly on each number (as relatively independent) in a cardinal setting, the very ability to do so, implies that we can implicitly extract it from an interdependent setting (where it exists in relation to other numbers).
However the considerable task still exists as to how to give this ordinal notion of number a precise mathematical identity (which we will look at closely in the next blog entry).