However, unlike Alberti and Leonardo, Dürer was most troubled by understanding not just the abstract notions of beauty but also as to how an artist can create beautiful images.

However, in more abstract and complicated CPUs and ISAs, a microprogram is often used to assist in translating instructions into various configuration signals for the CPU.

However, that would convey the idea of the physical body of people, whereas using the name of the language as the basis of the word gives it the more abstract connotation of a cultural sphere.

However, unlike some orthographies, English orthography often represents a very abstract underlying representation ( or morphophonemic form ) of English words.

However, leadership of West Germany saw reunification as a rather abstract goal.

However, production does not get carried out in the abstract, or by entering into arbitrary or random relations chosen at will.

However, because these tools do not implement the " web of abstract concepts " hiding behind the system of natural-language macros, or provide an ability to change the order of the source code from a machine-imposed sequence to one convenient to the human mind, they cannot properly be called literate programming tools in the sense intended by Knuth.

However, every experiment to date has shown these values to be proportional, and this proportionality gives rise to the abstract concept of mass.

However, the abstract formalization of the group axioms, detached as it is from the concrete nature of any particular group and its operation, allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way, while retaining their essential structural aspects.

However, some versions of nominalism hold that some particulars are abstract entities ( e. g. numbers ), while others are concrete entities — entities that do exist in space and time ( e. g. tables, chairs ).

However, the abstract elements of God ( goodness, wisdom, etc.

However, universals are not necessarily abstract and particulars are not necessarily concrete.

However, it has no readily-discernible timeframe, and combat is extremely abstract, leading many to not consider it as an actual wargame, or only tangentially as one.

However, the general definition remains valid in the more abstract setting of order theory where arbitrary partially ordered sets are considered.

However, the factory only returns an abstract pointer to the created concrete object.

However, the type of a concrete object ( and hence a concrete factory ) is known by the abstract factory ; for instance, the factory may read it from a configuration file.

However, it rejected the notion that all meaning comes from signs existing purely in the abstract space of langue.

However, Plato himself and Greeks had copied 1, 500 older Egyptian fraction abstract unities, one being a hekat unity scaled to ( 64 / 64 ) in the Akhmim Wooden Tablet, thereby not getting lost in fractions.

However, when concerning philosophic or ethic value, an object may be both a physical object and an abstract object ( e. g. an action ).

However, the formula, abstract-negative-concrete, suggests a flaw, or perhaps an incomplete-ness, in any initial thesis — it is too abstract and lacks the negative of trial, error and experience.

However, some abstract phenomena like " fun " and " hope " have properties which make it difficult to refer to them with a count noun.

However, many games are commonly classed as abstract strategy games which do not meet these criteria.

However, attempts have been made to argue that predicates are also ontologically committing, and thus that subject-predicate sentences bear additional ontological commitment to abstract objects such as universals, sets, or classes.

However, there is no exact, universally agreed, definition of the term " discrete mathematics.

However, it seems that many of the methods for solving linear and quadratic equations used by Diophantus go back to Babylonian mathematics.

However, Valla added the translation of ancient Greek works on mathematics ( firstly by Archimedes ), newly discovered and translated.

However, in the 1930s Gödel's incompleteness theorems convinced many mathematicians that mathematics cannot be reduced to logic alone, and Karl Popper concluded that " most mathematical theories are, like those of physics and biology, hypothetico-deductive: pure mathematics therefore turns out to be much closer to the natural sciences whose hypotheses are conjectures, than it seemed even recently.

However, among academic mathematics, the majority of mathematical papers published in the United States are written by academics outside of mathematics departments.

However, because the intuitionistic notion of truth is more restrictive than that of classical mathematics, the intuitionist must reject some assumptions of classical logic to ensure that everything he proves is in fact intuitionistically true.

" However, proponents of more limited forms of constructive mathematics would not assert that ZF itself is a constructive system.

However, it is currently an interdisciplinary science that collaborates with other fields such as chemistry, computer science, engineering, linguistics, mathematics, medicine and allied disciplines, philosophy, physics, and psychology.

However this definition does not allow star polytopes with interior structures, and so is restricted to certain areas of mathematics.

However, Pythagoras believed that the mathematics of music should be based on the specific ratio of 3: 2 whereas Ptolemy merely believed that it should just generally involve tetrachords and octaves.

However, for many Muslims there is no distinction ; all forms of art, the natural world, mathematics and science are all creations of God and therefore are reflections of the same thing-that is, God's will expressed through His Creation.

However, some finitist philosophers of mathematics and constructivists object to the notion.

However, with some luck, Smale was accepted as a graduate student at the University of Michigan's mathematics department.

However, this criticism is rejected by supporters of the social constructivist perspective because it misses the point that the very objects of mathematics are social constructs.

However, Daniel refused, because he wanted to study mathematics.

However, many prominent economists come from a background in mathematics, business, political science, law, sociology, or history.

However, he was unable to complete the curriculum's rigorous mathematics requirement and was asked to repeat his sophomore year.

However, as the mid dot was already in common use in the mathematics world to indicate multiplication, the SI rejected its use as the decimal mark.

However these researches are not philosophies of mathematics but rather the study of restricted forms of reasoning similar to Reverse Mathematics.

However, it is actually a broad, interdisciplinary field, incorporating not only aspects of computer science, but often diverse fields such as archival science, cognitive science, commerce, communications, law, library science, museology, management, mathematics, philosophy, public policy, and the social sciences.

However, the Weil conjectures in their scope are more like a single Hilbert problem, and Weil never intended them as a programme for all mathematics.

However, the link between the Riemann hypothesis and the Prime Number Theorem had been known before in Continental Europe, and Littlewood also wrote later in his book A mathematician ’ s miscellany that his actually only rediscovered result did not shed a bright light on the isolated state of British mathematics at the time.