Mathematicians often picture their science as entirely abstract and unrelated to reality. They hold that the development of mathematics follows its own ways, free from any practical needs, with no concern for the possible applications. Ultimately, the principles of mathematical reasoning are deemed to be eternal and non-mutable, prescribed once and forever by no matter who. Inebriated by the obvious success of formal methods in science and engineering, mathematicians are apt to believe that their science is to provide supreme (absolute) knowledge and the final criteria of consistency and truth.
Such is the body of common delusions about mathematics shared by many people who are not trained enough to observe their utter absurdity. Wisdom is difficult to learn, it has to be gradually acquired through persistent effort throughout one's life; studies like that are not much in favor today, as the leading economies are based on the division of labor. Professional philosophers lack scientific education; their chancy erudition can hardly become the right soil to raise universal ideas; moreover, modern professionals have often to be engaged in anything but their professional activities. The common public has almost no opportunity (and is not intended, nor encouraged to go in) for extensive reading; people remain widely ignorant, and they have no other choice as to trust to those who look authoritative enough. Thus, the present authorities feed us the authority of math.
However, a closer look at the activity of a mathematician will immediately reveal its overall resemblance to any other science, to the degree of a close kinship. Indeed, given a number of properties common to a class of objects, one normally tries to establish a number of interdependences, considering the possibility of prediction on the basis of a few already established facts. This is exactly what scientists do in physics, in chemistry, in biology, in linguistics, in psychology, or in history. This also pertains to learning, acquiring professional skills in any specific area, like medicine, storytelling, or plumbing. In any case, the regularities thus discovered are in no way arbitrary, they reflect the real order of things and the current modes of their usage. The only difference is in the nature of the objects and the domain of applicability.
That is, to comprehend the nature of mathematics, we need to grasp at least some idea of its object area. The details might come in the course of further study; but the general direction is to be chosen from the very beginning; otherwise, the activity just could not start.
Of course, indicating the object of mathematics is a nontrivial task, and hence all the controversy about the foundations of mathematics, and the range of common illusions. The situation is aggravated by the fact that mathematical ideas have never stopped developing, both in their wording and in their content, and it may be difficult to trace the origin of the currently recognizable fundamental blocks back to the roots of the trade.
Thus, today, we understand that there is something in common between 3 apples and 3 stones. With a little more mental effort we can admit that a collection of 2 stones and 1 apple is, in a way, like 3 stones, or 3 apples, and so is the collection of 1 stone and 2 apples, or even 1 stone, 1 apple, and 1 bird. With more experience in abstraction, we can also discover that 3 days, or 3 wishes, somehow fall in the same category. In practice, such associations are established through comparison of various collections of things (or just ideas) with the same reference collection (for example, 3 fingers). The final step is to remove (abstract from) any reference collection at all and speak about the number 3 as a common characteristic of any group of 3 distinct entities, regardless of their distinctions. This approach, stressing the universal commonality, is called quantitative.
Alternatively, one could focus on the very act of distinction, the individuality of things, treating them as qualitatively different, and hence incomparable. Obviously, this absolute distinction means as absolute equality, since the very impossibility of comparison prevents us from telling the difference as well as from establishing any commonality. That is, we can count unique things as indistinguishable, and hence identical. This gives us yet another mathematical primitive, an item, just something to count (a point, an element, an operation, a link, a value). In particular, we can count numbers.
The quantitative and qualitative aspects are intrinsically related to each other, they always go together in any practical act. Speaking of commonality, we mean distinction; speaking of distinction, we mean identity. We consider quantitative grades within the same quality, as well as the quantitative limits pertaining to a specific quality. This means that every individual thing must also be characterized in yet another way, allowing us to distinguish quality from quantity and put them in the same context. It is only in respect to this common base that some aspects of the whole could be called qualitative, while some other aspects would provide a qualitative assessment. In philosophy, this unity of quality and quantity is known as measure (not to confuse with the narrow mathematical term). Assigning something a measure is an act of measurement, in the most general sense.
For measurement, it is important that different things can be considered as equivalent, to a certain degree, that is, commeasurable. This possibility is primarily related to the very nature of human activity, which always takes some object to produce some product. The product could be called a material implementation (realization) of measure, since any objects within the same activity become comparable in respect to its product: they are either fit for production, or irrelevant, with the whole hierarchy of the possible grades of applicability.
Quality and quantity are the two complementary aspects of measure, and hence their distinction is relative. The same measure may compare things in a different respect, without changing the sense of comparison. This observation is almost trivial, since the result of any measurement (quantity) entirely depends on its method (quality). The same work can be done in many ways. Depending on how we distinguish countable things, we will obtain different counts.
The universality of the subject means that every two distinct things in the world can virtually be compared; however, any comparison needs an appropriate measure, which has to be established in practical activity, and hence the scope of actual comparability is determined by the current level of cultural development. That is, the very thought of comparison already means that there is an appropriate social background, and no idea can just enter one's head without a cultural instigation. Of course, there is no need to track any mathematical construct at all back to material production, since every activity, within a cultural environment, can become an industry on its own, representing some higher-level cultural regularities (reflexive activity). In particular, the product performing the role of a measure does not need to be a palpable thing; sometimes, it may be just an intricate interrelation almost impossible to embody or express. Nevertheless, the cultural determination of formal operations cannot be eliminated, even in the most abstract areas of science.
Within the same measure, the objects in the domain of commensurability are, in a sense, interchangeable, just like trade articles in the market. Each of them can be taken for a unit, with the other objects somehow related to it. This specific implementation of measurement is certainly not unique, since any other object can serve as a unit as well, producing its own scale, so that anything could be associated with a definite grade of the scale, which provides a common measure for all the objects of the same degree. Any measure will thus produce a hierarchy of scales, which is extremely mutable and multiform, each particular representation being a hierarchical structure, virtually corresponding to a possible structure of the matrix activity. Such hierarchical structures are what could roughly be taken for the object of mathematics as a science. Each branch of mathematics refers to a class of typical structures, the different ways to organize people's activities.
As an area of activity, mathematical science can be structured in the same way; the ideas of an exceptional character of mathematical knowledge are largely due to this apparent reflexivity. Still, mathematics is not unique in that. Any science at all is reflexive, since its cultural existence is impossible without the feedback from its application area, and many sciences involve the processes they are going to describe. Can you imagine a physicist that would not be a physical system? Or a biologist that would not be an organism (or a community of organisms)? Economical science is obviously engaged in a sort of trade; psychology requires intellect and emotions; linguists communicate in many languages; geology develops on the planet Earth. The only difference is in the character of reflexivity: in mathematics, it often (but not always) becomes explicit and intentional.
In this context, one could picture measure as an instance of categorization: a scale provides a number of categories (grades of the scale), and each individual act of measurement is to put the result in a definite category. This is a most common operation in people's everyday life, and we don't even pay attention to its essentially cultural nature, the necessity of an embedding activity. Categorization is never trivial. It must historically (practically) form as a cultural pattern (a hierarchy of scales) on the basis of mere comparison (distinction) in respect to the product of some activity. This is the link between humans and animals; the latter can immediately assess the biological importance of a stimulus and take reflective action, but they do not develop any scales, except in hierarchically organized communities with dynamic function exchange (the prototypes of the human society). In other words, a category must objectively exist before we put anything in it; and this cannot be but social (ideal) existence. As soon as a category becomes "wired" in the decision maker, there is no real choice, and we can only speak about categorization in the metaphorical manner. The same holds for acquired "categories" established in individual learning and encoded in one's mental activity and neural patterns; however social by their origin, such "psychological appliances" do not much differ from bodily organs, and their rigidity is felt as a lack of responsibility and limited freedom. From the hierarchical viewpoint, this means too much preference for one hierarchical structure to the detriment of the others, the suspension of hierarchical conversion.
It is important that measurement (elementary or discriminative) is not indispensable in human activity, and, in many cases, people can avoid it. There are other modes of assessment, and the integrity of the human culture is only achieved in the interplay of all the possibilities. Thus, to tell whether all the members of the family have come for dinner, we don't need to count them; we can detect somebody's absence at a glance. Similarly, we don't need to count the buttons on a coat to find that one is missing. The shape of the pillow is of no importance as long as it does not disturb our sleep. And we don't care for the truth of somebody's words as long as our communication is only to confirm sympathy or disgust. Some ethnic groups never come to the general ideas of number, or shape, since they don't need them in any practical respect. Similarly, the quantitative aspects of reality are irrelevant to little children below some culturally determined age. Even in science, a bulk of factual or operational knowledge is primarily accumulated without too much bothering about formal distinctions; this is, for instance, how the method of contemporary mathematics has virtually been born. However, the philosophical categories 'quality', 'quantity' and 'measure' reflect the fundamental organization of the world (regardless of conscious activity) and are universally applicable, so that anything at all can (thought does not need to) be measured (compared, categorized, evaluated), in complement to all the other attitudes.
I dwell so much on the preliminaries, since the details of mathematical measurement can easily be recollected as soon as there is an understanding of their cultural determination. Any mathematician could do it almost in no time; a less trained person would additionally need refreshing one's school reminiscences. A working scientist is the most difficult person to persuade, just because of the many formal habits that have penetrated the very core of one's personality. So, let such people do what they are made for without too much caring about the foundations of mathematics, which would not certainly be the best application of their talents.
There is no restriction as to the type of scale to produce a mathematical measure. Any activity can produce various formal counterparts, depending on the way we unfold its hierarchy. The most fundamental mathematical notions are as subject to reassessment as any auxiliaries. In the above example with abstract numbers, one can easily observe numerous conceptual vulnerabilities. Thus, putting all kinds of items in the same row, we refer to some culturally established operation of counting; the measurement procedure involves labeling individual items with the marks on a standard scale (a number of reference objects taken in a fixed order). That is, counting assumes enumeration. Natural numbers as a mathematical construct provide an abstract scale that does not depend on the particular implementation; however, in any practical measurement, we have to choose an appropriate instantiation of that abstract scale, from fingers and abacus balls to neural patterns in the brain, people in a queue, or a programmatic iterator. No implementation is perfect, and it took the humanity quite a lot to come to relatively stable procedure of counting; however, no one can guarantee that these habitual operations won't fail in some exotic conditions, requiring a different method of measurement. Just think about the very common finger scale and admit that some people might have a different number of fingers, or no fingers at all.
Assuming the adequacy of the materialized scale (the counting instrument), we still face the problem of the order of counting, that is, the necessity to arrange the objects to count in a row, so that we could sequentially associate them with the marks of the scale, eventually exhausting the collection and coming to the latest scale mark, which is exactly what we need, the number of items. Normally, we don't much bother about that, since, in many practical cases, the result does not depend on the counting order. But, in general, we need yet another iterator, taking the objects from the counted collection one by one for us to be able to relate the next object to the next grade of the scale. This enumeration is not trivial; the result may depend on the sequence obtained. For instance, the members of the sequence can cling to each other if taken in a particular order, while showing no interaction in other arrangements. Take the sequences of letters that may or may not form words during measurements; if we (for some reasons) detect words as single entities, the outcome of measurement will essentially depend on the mode of enumeration. Yet another common possibility is related to finite objects that disintegrate after an objectively determined time interval (e.g. like in radioactive decay). If we count starting from the long-lived items, we risk overlooking their shorter-lived companions in the whole.
Continuing the time theme, one could observe that the result of the measurement also depends on the rate of enumeration, both for the item collection and the scale. If the item sequence is produced much faster than the typical scaling time, some items just cannot be counted. In the worst case, the scaling time may depend on the next item, since some items might be more difficult to cope due to their size, weight, mobility, or cultural dependencies. Alternatively, the scale might occasionally fire "spikes" in packets, so that the same item would be counted several times (which may, for instance, correspond to yet another mathematical primitive, a set with repeated elements, a "bag"). The "classical" activity of counting is therefore to proceed in an "adiabatic" manner, with enough time between successive counts for the two iterators (the item collection and the scale) to relax and restore the "standard" state before each step. Everybody who has ever had experience in computer programming (especially networking) knows that such a steady operation may often be not easy to achieve.
We naturally come to the conclusion that the mathematical notion of a number reflects a very special way of operation subject to numerous restrictions. However common in our current cultural environment, such operations may be utterly impossible in other cultures (or on the other levels of culture), where the very idea of a number would be inappropriate. Of course, this does not mean the impossibility of mathematics in general; some other mathematical measures might come quite handy.
A mathematician could object that their science should not care for feasibility, and mathematical theories could develop regardless of any application, just to explore the formal dependencies. However, mathematics is just a kind of activity, and it is bound to run into the same problems as anything else. The famous Gödel theorems provide a bright example of inappropriate arithmetic coding producing the illusion of commensurability where there is none. There are reasons to believe that any diagonal proof is, in fact, an indication of the inapplicability of the corresponding mathematical theory.
Just for illustration, let us look closer at the common distinction between ordinal and cardinal numbers. As indicated above, counting implies sequencing, and hence the numeric scale is used in the "ordinal" sense, as an order imposed on the item collection. Provided different orderings are irrelevant (and, in particular, they give the same item count), we could think of the number of items as a cardinal number, a characteristic of the collection as a whole, its "volume" (or "mass"). Formally expressed by the same mathematical constructs (numbers), such "bulk" measures reflect a different approach to measurement that does not need resolving individual elements within the whole. Instead of counting bricks in a pile, we could just weigh the pile and thus estimate the number of bricks in an indirect way, knowing the weight of a brick. Similarly, we could measure the length of the border built of the bricks from the pile; and, again, the number of bricks can be formally estimated knowing the size of a brick. Obviously, a bulk measure does not necessarily allow quantitative judgment on the elements of the collection; thus, in a heap of stones (instead of standard bricks), individual stones may be very different in size or mass, so that the weight or dimensions of the heap do not provide any information on its structure.
In physics, we also distinguish intensive (like temperature, pressure, or entropy) and extensive (like mass and volume) quantities: the former are evaluated for the whole system; the latter sum up from the corresponding values for its parts. There are obvious analogs of intensive measures in mathematics: the dimensionality of a space, the topological index, the momenta of a statistical distribution, or algorithmic complexity. Even if we can subdivide a mathematical construct into a series of smaller constructs of the same kind, the estimates of intensive quantities for the parts won't trivially sum up into the corresponding value for the whole. For instance, if we split a rectangle into several (non-intersecting) rectangles, the ratios of the side lengths for individual components have nothing in common with the same ratio for the original (compare with the areas of the parts that sum up to the whole area); one could also find intermediate cases: thus, the sum of the perimeters of the components rectangles does not equal the perimeter of the whole, but the two values can be correlated for some types of partitioning.
Intensive characteristics are related to the mathematical idea of a shape, as complementary to that of a number. The same considerations about identity and distinction apply here. One could roughly describe things as "round", "square", "jagged" etc. We distinguish smoothness from raggedness, chaos from order, similarity from distinction, or failure from success... In all such cases, we take different things and treat (employ) them in a similar manner. This makes these things virtually identical (and hence subject to quantitative description), but also different from other things that cannot be used the same way.
While a number is to stress the quantitative aspects of measure, shape mainly refers to quality. We can describe shapes with numbers; but such descriptions do not convey the integrity of shape, they only explain why we perceive certain things as different shapes. In many practical situation (in the zones of cultural stability), we can guess shapes from a selection of numerical parameters. However, this does not reduce shapes to numbers, since the same shape could arise under quite different conditions, where typical numerical expressions are no longer applicable. Conversely, the same collection of numbers may refer to an entirely different shape; we can visualize one shape with another, but this won't make them qualitatively equivalent. In other words, a shape is a higher-order entity respective to any particular parametrizations, and it is defined respect to other shapes rather than by any numerical expressions. On the other hand, a numerically parametrized shape can be considered as more abstract, since, among all the characteristics of the shape, it selects just a few; one could call such a parametrization "the shape of a shape". This is quite similar to how we represent a spatial point with its coordinates, thus making it an abstraction of a point.
Shapes do not exhaust the range of possible qualities. For instance, we feel certain commonality between two apples that makes them not exactly like, say, two hedgehogs. Moreover, there are different sorts of apples, and even apples of the same sort may be of different grades. Each individual thing is virtually different from any other thing, and that is why we can speak of any individuality at all. Still, within a definite measure, its qualitative aspect will characterized the commonality of individual things making them countable units. One could consider shapes as formal qualities (a "quantitative" quality) abstracted from any particular measure, just like a number as an abstraction of quantity. In this formal sense, apples and hedgehogs could be put in the same category (of approximately round things) and counted on the same footing. This is where mathematics is at its best.
Shapes and numbers could be compared to space and time in physics. In the same way, the items composing a shape can be ordered (counted) in a kind of trajectory, and conversely, a class of trajectories can be associated with a definite duration (a number). There is the same mutuality, the necessity of distinction and its inevitably relative character. And just like time is related to cyclic reproduction of the world (or any its part), the possibility of counting (and any numeric evaluation in general) is due to the repetitive actions and operations in human activity. In a way, mathematics could be called a model of cultural space-time, as a level of space and time in general.
Associating intensive measures with shapes, we come to a very general idea of shape applicable to abstract entities as well. For instance, mathematical theories (and their elements) can be related to some measure of truth, which is obviously an intensive parameter, so that the truth of the individual statements does not imply the truth of the whole theory. Here, exactly like in physics, the same measure is applied to the whole in a sense different from measuring its components, thus unfolding a hierarchy of measure. In particular, the hierarchy of truth determines the shape of a theory. Of course, the same theory can be related to other intensive measures (like decidability, productivity, predictive power etc.) and may have a different shape from yet another perspective. These "special" shapes are not entirely independent. In some cases, intensive parameters may become extensive, and vice versa. Thus, the mass of a compound particle builds up from the masses of individual components in nonrelativistic macroscopic systems; however, the mass of an atomic nucleus cannot be reduced to the masses of individual nucleons. A complex system may behave as a whole in one context, while undergoing mere shape changes in another. Moreover, the sequence of changes may essentially depend on the system's history. Similarly, a mathematical theory may link intensive measures to each other, thus making them "less intensive". In the context of some theory, two statements (assessed as true or false) can be combined in a compound statement (using some logical junctions), so that the truth value of the result could be derived from the truth values of the components. Different sets of junctions differently shape the theory (producing its equivalent formulations), and one of the primary concerns of a mathematician is to determine the range of constructs preserving the same overall structure, which is indeed not evident from the very beginning, being revealed in the course of development, like in other sciences.