Dear Sir,
We do not understand why mathematics should be made incomprehensible in stead of being scientific and rational.
Numbers are a property of all substances by which we distinguish between similars. Distinguishing between similars is a matter of perception at "here-now". When there is the perception of an object without similars, it is one. Depending upon the repetition of the experience of such perception, we assign a different number to each set of such perceptions, which we call the number sequence. (We are not elaborating the exact mechanism but leaving it for a different occasion). We can differentiate between similars only if the object has a fixed structure. Thus, we do not assign numbers to fluids, though we assign numbers to the comparison of the volume with a fixed or unit volume.
Number is not directly associated with the object, but is associated with whether there are similars or not. Thus, particles, that are a composite of sub-particles, exhibit these numbers differently, because similarities in the two cases are different. The number associated with a particle can repeat itself in the case of a sub-particle also depending upon whether there are similars or not. The rational numbers are only distinctions between perceptions of two sets of numbers. For example, if a particle consists of x number of similar sub-particles and if we take away y number of such sub-particles by differentiating them from other sub-particles, both operations are fully perceptible. Thus, we call the number a rational number. But where such distinction is blurred, we call it an irrational number.
Since all perceptions are quantized, the increase or decrease in number sequence take the shape of 1,2,3,4,5,....n and n,....5,4,3,2,1 respectively. When the increase or decrease is linear, we call the operations as addition or subtraction. When it is non-linear, we call the operations as multiplication and division. By non-linear we mean only partially similar. When an object does not exist at "here-now", we call the number associated with it as zero. This only implies the absence of the object or perception at "here-now" but not its perception from our time invariant memory. Since the object is not perceived at "here-now", no number can be associated with it. Hence all linear operations involving zero leaves the number associated with the object unchanged (the use of zero as a decimal function has a different explanation. We are leaving it for a different occasion).
In the case of multiplication, since it represents an operation involving another object and since one part of the combined operation does not exist at "here-now", the result of the entire operation cannot be perceived. Thus, the result of multiplication by zero is zero. In the case of non-linear reduction (division) by zero, the non-linear part that is not perceived at "here-now" is not perceived. Since it represents an operation involving another object at here-now and since the operative part does not exist at "here-now", the perception of the entire operation remains unchanged. Thus, the result of division by zero leaves the number associated with the object unchanged. However, in modern mathematics, it is wrongly associated with infinity.
Infinity is like one - there is the perception of an object without similars. But unlike one, the dimensions of the object are not fully perceived (we have discussed it elaborately in our essay). There cannot be an infinite set of numbers - it is only a very big number. Since perception of numbers is related to "here-now", and since perception of objects with infinite dimension are not possible at "here-now", all operations involving infinity is void.
The perception of: "numbers such as the square root of 2 which cannot be written as the ratio of integers", stand in a different footing. Squaring is a non-linear operation. Square root is a non-linear operation involving a field in two Dimensions, which has a second order number. You cannot take the square root of 2 bikes. But you can take the square root of a field measuring 2 square meters. While the dimensions of the two non-linear components of "square root of 2" are perceptible (such as 2m x 1m), their individual components after a specific operation involving non-linear reduction, may not be perceptible. However, since the field, both prior to and after the operation, exist at here-now, it has a number associated with it. Thus, we restrict our description of this number to the nearest perceptible fraction (components).
All operations are conducted by an agent who has the ability to indulge in such operation. In other words, it symbolizes a kind of "ownership" over the object of operation. This ownership is indicated by the sign preceding the number explicitly or implicitly. When such "ownership" does not exist, yet the object exists, the numbers associated with such objects are called negative numbers. This absence of ownership is indicated by the - sign preceding the number explicitly. When we talk about "integers (..., -3, -2, -1, 0, 1, 2, 3, ..., negative and positive, including zero)", we indicate this change of "ownership" pattern. Since it indicates a sequence, it is time variant. Operations involving numbers are carried out at "here-now". But no such operation is possible with the so-called integers. This distinguishes numbers from the so-called integers. The concept of "aleph-zero counting numbers and aleph-zero integers" is wholly erroneous.
Since rational numbers are only distinctions between perceptions of two sets of numbers, to say that: "The set of rational numbers is incomplete" is not a proper description of facts. In fact, it can be highly misleading. So is Ostrowski's theorem. There is no scientific basis for accepting his views. We need theories to explain reality by showing the correspondence between theoretical description and actual observation in a logically consistent manner. When the existing theories are sufficient to explain reality in a logically consistent manner, bringing in additional factors has only the effect of shifting away from reality. We need not assume that what we perceive is wrong and what we cannot perceive is right. This concept, couched in the language of incomprehensibility has been used by scientists to fool the gullible public for hundreds of years to lead a cozy life at public expenses. We can explain everything using ordinary mathematics derived from fundamental principles. A mature person cannot be a child, because the only difference in perception between them is that while the mature person has more "experience" (hence a bigger memory bank), the child lacks it or has it in a small measure. To invoke the inner child means to assume that one's memory is all wrong, which means one has gone mad.
The description of Piaget's method of assimilation and accommodation itself by Mr. Paster is wrong. Hence it is no wonder that the conclusions arrived at are misleading. For example: Mr. Paster has written: "Assimilation means that we take an observation or experience and add it to our existing conceptual structure, enhancing the structure that was already in place, but not transforming that structure into a new conceptual structure". This simply means that we "learn" through experience by replacing our earlier ideas with new ideas - making structural adjustments. This "learning" is nothing but our "new concept", because, with the same input different persons will add differently to their existing concepts making it a "new concept" every time. There is no other meaning for "add it to our existing conceptual structure".
He further writes: "Accommodation, on the other hand, means that the observation or experience has been so novel or discordant that we cannot absorb it into our existing conceptual structure. Instead, we must modify our conceptual structure, accommodating our world view to incorporate the latest novel, discordant event." This simply means the same thing as assimilation, except that the magnitude of the latest structural adjustment to our memory has been comparatively much bigger than usual. Admittedly, "Piaget considered himself an epistemologist first, drawing conclusions about the nature of knowledge from his observations of human cognitive development." The above description fits his views.
Mr. Paster has used P-adic mathematics to model Piaget's processes of assimilation and accommodation. He writes: "Assimilation means one of the digits of the prior p-adic number has gotten larger, but we still have the same number of digits". This is something beyond us. Firstly, we do not see the necessity of any other ad hoc mathematics beyond simple natural mathematics. Secondly, digit is nothing but the name assigned for a certain number of perception of similar objects. For example, if we had perception of 1, then another 1 similar objects, we assign this type of perception a name, which may be two. For every such repetition, we assign different names and call these digits. Since number is only one of the perceived properties of objects and not the object proper and since larger number implies addition of similar objects, we do not see how "the prior p-adic number" can get larger, but "still have the same number of digits".
He further writes: "Accommodation in its simplest form means that we have one more digit, one more level of hierarchy. Accomodation can also take other forms: A segment of the prior p-adic number can be preserved but encapsulated within a different enclosure. Or levels of enclosure can collapse into a larger single enclosure". This is an entirely wrong description of facts. If "we have one more digit", we do not get "one more level of hierarchy." Hierarchy implies "difference in class". But if "we have one more digit", we have one more object of the "same class". If a "segment of the prior p-adic number can be preserved but encapsulated within a different enclosure," then the "addition of one more digit" is not possible, unless it interacts with the enclosure. If it interacts, then the concept of enclosure is meaningless. The concept of "levels of enclosure can collapse into a larger single enclosure" means nothing but making structural adjustments, which is again nothing but assimilation as explained earlier.
We agree with Piaget's concept of equilibration as the driving force, but we interpret it differently and much more universally. We leave it for the time being. Regarding minimum length, we had discussed elaborately in our essay. Regarding "most mainstream physicists reject out of hand a role for physics in explaining the mind" all we can say is "sour grapes". Since they have not understood the concept, they say so. As we have described repeatedly, we explain mental functions mechanically. We accept thought as the inertia of mind. Regarding dimension, we have written in various threads in this competition to show the nature of dimension, what the ten spatial dimensions are, and why time cannot be a dimension.
Finally, most of the "scientific" terms are nothing but mere words to show off one's "knowledge" through the cult of incomprehensibility. This is unfortunate, but true.
Regards,
basudeba
