Number

Basic calculations

In this model what is to be determined as number is already some whole, on which there is possibility to apply certain differentiations. So we say that something is "two" if we can apply the differentiation which would determine it as two. It is easy to imagine such possibility, as possibility to divide the whole, after which we can apply one and the same differentiation to both parts.
That I'm using "divide" and "both" here, shouldn't be seen as a problem of self-referencing in definition of number, because what I'm doing here is try to describe the sub-conceptual things (as differentiations) by using words, and it is just because of that I can't avoid using the concepts whose possibility I'm trying to describe.

On other side the resulting parts each by itself, are the concept of one, but just in the context of the concept of pair, just because they were those which were determined as same. Of course that which was determined as one, can also be determined as pair by further determination if there is possibility to apply the "pair" differentiation tree on it.

Having the faculty of abstract imagination which is faculty to imagine the possibility for application of certain concept on some possible experience, we can imagine the experience where we have concept of one, and then we can imagine that in same experience we have the same concept just in different place. Still in the same experience we are aware that the differentiation of "pair" is applicable, making the same experience "pair", or in other words we are aware that in every possible experience where we can apply differentiation tree of "one and one more" (as described), we can also apply differentiation tree of two, or pair.  In such way we are aware that in every possible experience "1+1=2".  (one and one more is two).
That the same differentiations and "truths" are applicable on apples, bananas or whatever is not important, as the differentiations which make those truths come from the faculty of mind alone, hence can't be named empirical truths - and when applying those differentiations actually what makes apple-apple its concept is actually abstracted from the truth, the truth of "1+1=2" and its meaning is same whether applied to apples, bananas or houses.

 Same thing with little recursion (applying same differentiation on already applied differentiations which is possible in this model) can be used to determine 3, 4, and bigger  numbers, although it might be even possible that we have innate differentiations for pair, triplet, etc..., and are using the recursive differentiation just for bigger numbers (e.g. 13).
It should be pointed that same as with previous propositions, we shouldn't think that there are really two sides ("1+1" and "2"), which are connected in some magical way, it is all one experience, in which the mind is aware of applicability of the both sides of the propositions.

In bigger numbers, which can’t be held in mind as those smaller can, and as such can’t be even seen as concepts, the judgment that 120+12=132 can again be seen in designation of 132 as possibility to separate 132 parts, and then separating of this quantity (as we have this might of separation by having might of differentiation) to two quantities in some particular way, in the example above to one group of 12, and one group of 120. (Of course we can’t do this naturally; we never can imagine 132 things for starters, in this case we use additional method for adding decade numbers learned in school.)

Multiplying , e.g. 2 times 3, is applying of differentiation to a differentiation, separating something into 2 groups , where in each of those groups there are 3 things. It might be noted that what I’m taking is actually division, or in the earlier case when I talked about addition – subtraction, but here it is about the awareness of the mathematical fact, about the final judgment where mind sees the whole of the judgment “2*3=6” or “2+3=5” in its necessity, and that can only be when the mind is starting with the whole, and then applying the differentiations on it, finding the “parts” in it. So in case of 2*3=6, it can imagine (or see) 6 things, and then separate them on two sets, each having three things. Or in case of awareness of “2+3=5” truth, it can start with the 5 things and then separate them on two groups of 2 and 3 things respectively. Of course there might be other (more complicated) ways to be aware of 2+3=5, but they would just be some more complex wholes, and more complex applications of the differentiations – for example in case of 1+2=3 one can differentiate over the whole method of imagining 1 and keeping it under concept of one, and then imagining a pair (two imagined dots for example under a concept of two), and then again applying new (parallel) differentiation on that cognitive context, in which the triplet would be determined.

But let’s continue… defined as such, multiplication is implicitly present in Arabic number system (normal decade system). There, number 23 in fact means that we can do separation of two 10 (2*10) and one 3 from the number imagined.

Even at first the fractions can be seen as something which gets outside of whole numbers, e.g. it would be normal to say that 3/4ths is not whole number, if we analyze how we comprehend this number it gets clear that it is again the possibility to divide, but now it is imagine that what we call “one”, in fact can be divided to four parts – the fourths. So in that way 3/4 is 3 * 1/4, where fourths are which are seen as fundamental parts, but it also carry the fact that 4 such parts we take for one. So in that way it is again just about agreement on how to group things, and what would be called as one, what would be seen as parts of the one, and what would be seen as groups of the one.

Different Representations of Number

Those same powers to play with grouping and dividing, is what makes possible the concept of decimal numbers, where from the very start it is supposed that what we call one, is in fact thing which consist of ten separate parts (can be divided to 10 separate parts). Those parts themselves in turn can be separated to 10 parts etc... If we become aware of that, we see that any decimal number is not different from decade numbers, just with additional sign (decimal point), with which we mark the place of “ones” -  left  of the ones are the groups (tens, hundreds, etc…), and right of the sign are what we define as parts.

So number 24.3 does not mean anything else but that we have 243 parts, but with the additional difference that we are interested in groups of ten of those parts to be seen as ones.

When we try to convert fractions into decimal numbers, we are just doing regrouping - trying to find out how much groups we will have if we group all those basic ones into groups of x, or groups of y parts. So 4/5 means that we count the fifths of the whole and 0.8 means we count the 10ths of the whole, which could be stated as 8/10. 

Here we get to the problem of “incompatibility” of different possibilities of separations, because it is easy to imagine something divided to 10 parts, and then group those parts two by two getting to 5 sets (1/5=0.2), and hence it is possible to say 1.6=8/5, but we get to problems when we try to regroup 1/3 or something that is in fact group of three things, in group of tens. So 1/3 can’t be just stated as number in base 10. If we try to convert the 1/3 to a decimal number, and start dividing, first we see that if we say that 1 is in fact whole of 10 parts (implicit in decimal numbers), we can divide 10 on 3 groups of 3 parts, and 1 part would be left which should be divided next, so 10/3=3+1/3, or by regrouping (moving the floating point) 1/3=0.3 + 0.1/3. If we want to go on with division, in decimal numbers system we get to 1/3=0.3 + 0.03 + 0.01/3=0.33 + 0.01/3. Further we get to 1/3=0.333 + 0.001/3 and so on…

It is this principle of dividing (this method), that we are representing further with 0.33(3) (3 repeating), and adding of 3*10^-n to infinity is a consequence not of the part which is already divided (0.333...), but of what is left to be divided 0.000----01/3. So, it can be seen as 1/3 where we allow dividing 1 on three parts (1 is 3 thirds), or it can be seen as 0.33(3) if we allow 1 to be divided just on 10 parts (1 is 10 tenths). So 0.3(3) in fact represents the set of results we are getting in the process of recursive dividing of the starting 1 on 3 parts, that is 0.3(3)=1/3, 0.3(3)=0.3+0.1/3 and so on, so the representation 0.3(3) is actually short-hand for describing any process, through which we get to determine the never ending sequence of cipher 3.

Following that, we can say that 3*0.33(3) is in fact 1/3*3, or 3*0.33 + 0.01/3 = 0.99 + 0.01 = 1 (after 2 divisions), but also 0.99(9) – of course 1=0.9 + 1/10 , then 1=0.99 + 0.1/10, which would mean that as a result of this kind of dividing we can say that 1=0.99(9), but it would be something like saying: you can divide 10 parts to 10 people if you divide 9 parts between them (as each of those 9 parts are imagined divisible by 10 being decimal) and then divide the remaining part between them – even it is true, nobody would divide 10 parts to 10 people in such way. So 1=0.99(9), but it is so just in the way explained above.

There is nothing magical here, but we must be aware of important thing that we can’t expect that what can be grouped in one way, or what have principle of dividing in one way, be possible to be regrouped or have possibility of dividing in other way, unless we add new rules to the game - if we see 0.33(3) not as finished “counted” quantity (of some infinitely small quanta, as its value would be never finished as simple number), but only as description of our method to get to that result by using a recursive dividing. This analysis helped us to see that the counting is just a consequence of possibility of dividing, and that the dividing is and stays the only “real” basis of the concept of numbers (at least if we don’t want to have problems in mathematics with that concept).

Also here we see that the “infinity” which appears in the concept of 1/3 when turned in decade system, is just infinity as impossibility to finish some recursive process of division under certain rules, and that 0.33(3)  which represents 1/3^rd is NOT some infinite sum of infinitesimals, but is just a method of division, in which the (3) at then end speaks about the method(s) used, which can then be used to get again to definite result when doing some other operation (like multiplying by three).

So there is no real infinity there which is then somehow applicable in the mathematical operations, the infinity or repeating to infinity is just a consequence of the method of dividing, and the method of dividing is what holds its certainty and can only get us to some certain result when doing calculations.

Or to put it in more formal way:

1/3= (10/3)/10=(3+1/3)/10=0.3+ (1/3)/10

The first part or 0.3 is one part of the solution, but the second part (1/3)/10 as we can see again is a basis for new iteration of the division with which we started just divided by 10. So by doing it again and again, we continue dividing the rest, getting new results we add to the sum (0.03, 0.003 etc…), but always that second part stays, which forces us to apply new and new iteration in the infinity. So again… the infinity here comes just from the way we are doing division – trying to divide 1/3 in terms of tenths, the infinity is not present in the 1/3 – it is fully determined number.

But if we don’t accept concept of the number as dividing, but instead look at number as counting – so we start from parts to get to the whole, we will get to such non-senses as infinities etc…

Awareness of possibility to count is based on the possibility to differentiate. For counting is nothing more then iterating through the individual “parts” of certain multitude of things. It basically requires differentiation of things on two sets, the ones we have already iterated over (counted), and the one that are left to count – if we don’t remember which elements we already iterated over, we can’t continue our counting. Of course when counting we get the things one by one from the one group, map then to the simultaneous iteration over the natural numbers (1, 2, 3…) (we see here that counting already needs the natural numbers as given) and put them in the other group of already counted items. It is of course wiser that we can use certain way of iteration so we don’t have to remember the whole set that we already iterated over, where we remember just the last one we iterated over (e.g. last planet we counted), and which is next one we can determine by: a).the law of our iteration, and b)the last one we iterated over. By doing so we implicitly again divide the things on two groups, only that we don’t have to keep already iterated ones in memory. The law of iteration could be counting from closest to Sun to most distant to Sun in case of planets around Solar system (or by their mass, or by whatever), or top to bottom, left to right counting in case of a dots on a paper. To reiterate… such counting if seen as the basis of the numbers produces such needs as need for infinities (infinite counting, infinitely small parts etc…) - what is needed is to see counting for what it is, a consequence of the possibility of dividing, and mapping things to already learned natural numbers, and instead to connect the concept of number to dividing, approach which if used removes the need of infinities, which would further be argued.

Series

In similar way to the way the cipher 3 was reappearing because of a certain method of dividing, any infinite series can’t just appear from nowhere. It must be generated by some recursive method of dividing of one original whole, a recursive method which is impossible to finish. Such method of dividing is what carries the n^th article of those series but also the rest that must be divided by the same method. In that way, the sum of some series presented by a definition of the n­^th article in the series must not be seen as to be result of summing the articles if it is done until end (as nobody will claim that he can do infinite sums), but it is sum as a starting whole, which divided by some recursive method is used to generate the articles of the series.

So, 1+a+a²+…=1/(1-a) (for a<1) doesn’t mean that you would reach the finite sum of 1/(1-a) when you succeed adding the infinite numbers of elements, but that there is a recursive method of dividing 1 on (1-a)  parts (details will be analyzed little further) which generates the series 1+a+a²+…, but also which generates the rest to be divided. So, when we have this series on mind, we must also have on mind not just the first part of the infinity, but the “power” that generates this series in infinity – the rest which is yet to be divided. It is just with such image in mind that we can comprehend the sum of infinity, not as a sum of infinite number of elements, but to see the sum as finding the method which generates those never-ending series from the starting sum.

Formally:

As a<1 we can say that a=1/q where q is natural number.

Then if we try to divide the A=1/(1-a)=[1/(1-1/q)] in terms of q, we get…

[1/(1-1/q)]=[q/(1-1/q)]/q=[q^2/(q-1)]/q=[(q^2-q+q)/(q-1)]/q=
[{(q-1)*q+q}/(q-1)]/q=[q+1/(1-1/q)]/q=1+[1/(1-1/q)]/q=1+A/q

The result of course in terms of a, would be 1+[1/(1-a)]*a, but that notion doesn’t show us the issue so clearly, as it is in terms of rational numbers (a<1), and here we are arguing that the basis of the numbers is in the end the division done by whole numbers (meaning that we can comprehend just such dividing). It should be noted though that the division with (1-1/q) doesn’t make sense, but the ratio which we can imagine is [1/(1-1/q)]=q/(q-1), where we assume that we have q ones which are divisible on (q-1) parts (both whole numbers).

Anyway, we see that we got to recursive formula, same as we did when we tried to show 1/3^rd in terms of tenths. Here we see again that it is not that somebody can sum infinite sequence of numbers, but that the “power” which generates series is the recursive method which tries to restate one ratio in terms of other factor.

Limits

This stays when we are trying to find “limit of certain function, when the argument gets closer to infinity”, as it is usually expressed. I will try to express this without using the vague term of infinity, instead using the notion of number as it was defined until now, as something which can be divided by some method, and I hope it would show that  what is expressed by those terms as infinity is in fact the method which is used to divide one “whole”, but which because of the nature of that method can’t stop, but instead is producing iterations again and again, which we can call infinity, but which is just a sign that the method can’t end by dividing one whole in certain way.

Let’s take for example: limit(n->inf) of 1/aⁿ=0

The common sense is that because n goes to infinity, the value of 1/aⁿ gets closer and closer to 0. This reasoning is as vague as saying that 0.3(3) gets closer to 1/3 and that if we repeat the cipher 3 in infinity we will in fact get 1/3^rd.

What will we do here, is we will take 1/aⁿ and transform it to a partial sum of some series. Doing so we will get a formula for elements of the series, and will show that the result (0) can be taken as starting whole of a recursive method of dividing which will generate the articles of that series.

For making it easier we transform the limit to the equivalent:

Limit_n->inf(1-1/aⁿ)=1 , and we use:

1-1/a^N= sum_1toN((a-1)/aⁿ)

So, 1-1/a^N is a partial sum of the series of (a-1)/aⁿ.

Now what is left is to show that the definite value of the limit (1 instead of 0 in the restated equivalent limit) is in fact a number, from which by certain recursive method we can get the series with elements (a-1)/aⁿ.

And that is trivial:

1=(a)/a=(a-1+1)/a=[(a-1)/a]+(1)/a

We see here that we get the first part which is the element of the series, and that we get the rest 1/a containing a 1, on which we can iterate again, and produce the next elements from the series (a-1)/aⁿ.

Now if we see the whole of the process we’ve done, we get that:

a)      1 can be divided by recursive process which results in the series (a-1)/aⁿ

b)      The partial sum of those series is 1-1/a^N

c)      By 1-1/a^N is result of dividing to certain point, but also there is the rest to be divided

So we can now interpret limit as: the result of the limit, can be divided by certain recursive method, will produce series whose partial sum will be the formula which is under the limit. So it is not that if we sum the series to infinity (some vague sense of turning the partial sum into sum of the whole of series), but that the certain partial sum will be generated by the method of recursive dividing of that starting whole, and the limit is really stating this fact about this particular recursive method of dividing, not about the partial sum, not about the series.

Side note: If we analyze the way we are doing dividing here it shows that the case of 1=0.99(9) was just special case of this limit if we change a=10, so for the concrete case we would say that following things mark same equation: 1=1 ; 1=0.99(9) ; lim_n->inf1/10ⁿ =0, just seen in different ways.

In some cases we would require parameterized recursion method to get to the series – a one “variable” which is sent to the next iterative step.

Such would be the case with limit_n->inf­(1/n)=0

In this case we can reformulate this with limit_n->inf(1/n-1)=-1

If we take the function under limit as a partial sum, we get that it is partial sum of the 1/(n+1)-1/n, or -1/n(n+1), so we should now show that the series -1/n(n+1) can be got by developing -1 by some recursive method.

And it is trivial as we shouldn’t go further then the equation we used for getting the element of the series, that is:

-1/n=-[1/n(n+1)]+(-1)/(n+1)

We see there that by changing n=1 for the first iteration, we get the start of recursion that divides -1/1=-1 to a particular series, as we get -1/n(n+1) for the current iteration and we get the rest of -1/(n+1), which would clearly give us -1/(n+1)(n+2) in the next iteration etc…, exactly the series whose partial sum is the formula under the limit.

So the limit_n->inf(1/n)=0 doesn’t mean that 1/n will get to 0 ever, but that 1/n-1 is a partial sum of a series of type -1/n(n+1), and that those series can be generated from -1.

Hence neither the series nor the limit of their partial sum can ever get to some final result, but they are product of the certain method of dividing of -1, and just by finding out that how this series or partial sums are generated, that we can deduce what were they result of, and to state the equation: limit_n->inf(1/n)=0, where limits addresses not the partial sum, but the recursive dividing which can generate it.

 Again “infinities” doesn’t have anything to do with it, other then the fact that the recursive method is such that never ends the dividing, hence one can’t finish dividing by that certain method.

Integration

Similarly to what we’ve done in the analyzing of the limit, we can analyze an integral. Namely to show that we can speak of integral without mentioning any infinitesimals, nor any kinds of other infinities (other then infinity as impossibility to finish a recursive method of dividing), we should show that what is seen as a result of the integration can be divided recursively in such a way, that we get series whose partial sums have forms which correspond to our usual understanding of the term integration.

Let’s take for example the integral of x, which turns to be 1/2x².

In the usual meaning of the integral as infinite sum of infinitesimals, we can probably represent it with the limit of the sum, when n tends to infinity:

sum(i=1..x) of x/n * x*i/n

This sum is equal to x²(n+1)/2n. Now, as we did with the limit we try to find out to which series this result is the partial sum.

Finding the difference of the sums for n+1 and n, we get that this is partial sum of the series with element with form: 1/2n(n+1) .

And to get to this type of series we can use this recursive dividing method:

(1/2)/n=1/2n(n+1)+(1/2)/(n+1), where we start the dividing with n=1, that is ½.

So we see that ½, can be divided by a recursive method which gives a series whose partial sum when put under limit correspond to our usual notion of the integral.

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