Once again I am going to illustrate 4 distinct meanings of number - illustrating with respect to the number 3 - before then showing that are in all inextricably linked in experience.

1) We start with the standard cardinal notion of 3 which represents the accepted quantitative notion of number e.g. 3 cups on a table. Number here is given an analytic independent identity (without qualitative distinction).

2) Here we have a very distinctive notion of 3 as now collectively applying to all groups (containing 3 members). So 3 can apply to 1, 2, 3, 4,.... groups without limit.

Now enormous confusion exists in Mathematics with both 1) and 2) generally confused with each other.

This is a crucially important point as the proper understanding of multiplication depends on this distinction.

Now again using 3 in the first sense might notice 3 cups and later - say - 3 letters in the hallway and perhaps then 3 cars in the driveway.

However strictly the recognition of 3 in each case would necessarily remain independent of each other.

So therefore the crucial factor in being able to establish a connection as between each group is the recognition that 3 now plays - as in 2) - a collective role (i.e. as what is common to each group).

As I say these two meanings with respect to number are intimately tied up with the process of multiplication.

Imagine two rows of coins laid out in rectangular fashion with 3 coins in each row.

Now from a multiplicative perspective we would represent this as 3 * 2.

So what is involved here is the initial recognition of 3 coins (in each row) in an independent manner.

Now if we only recognised the notion of 3 as independent as in 1) then we could only represent the total number of coins in an additive manner as 3 + 3 (where again both are interpreted in an independent manner).

However multiplication requires that we likewise recognise 3 as interdependent in a collective sense. This thereby enables us to see the common relationship as between each row.

Therefore in the relationship 3 * 2, 2 now indicates the common rather than independent notion of 2.

In this sense multiplication necessarily entails both 1) and 2) with respect to the interpretation of number with the first number relating to 1) and all subsequent multipliers to 2).

However we also have two further meanings of 3.

3) in this case 3 represents an individual group where the 3 members are all related to each other in an ordinal manner. This is the corresponding qualitative notion of number where the members of the group are identified as 1st, 2nd and 3rd respectively.

So with 1) 3 = 1 + 1 + 1; however with 3) 3 = 1st,+ 2nd + 3rd

4) In this case 3 no takes on a collective meaning where again it refers to the common recognition of all groups of 3 (where each is defined in a qualitative ordinal manner).

Thus in its simplest terms 4) represents at an ordinal level what 2) was earlier seen to represent at a cardinal level.

Now in dynamic experiential terms, it is impossible to separate these meanings, for they all imply each other in a complementary manner.

Thus the recognition of 3 explicitly in a cardinal manner implies the corresponding implicit recognition of 3 in an ordinal fashion.

Equally in reverse the explicit recognition of 3 in ordinal terms, implies the corresponding implicit recognition of 3 in a cardinal manner.

Likewise in a similar fashion, explicit recognition in individual terms implies implicit recognition in collective terms and explicit recognition in collective terms implies implicit recognition in an individual manner.

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