Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system.

Although Scheler later criticised Husserl's idealistic logical approach and proposed instead a " phenomenology of love ", he states that he remained " deeply indebted " to Husserl throughout his work.

Although the logical consequence relation is only semidecidable, much progress has been made in automated theorem proving in first-order logic.

Although his work was the first of its kind, it was unknown to the larger logical community.

Although biologists consider it contrary to the evidence of surviving Gondwanan lineages, geologists consider the logical possibility that Zealandia may have been completely submerged about 23 million years ago.

Although Descartes did not pursue the subject, he preceded Leibniz in envisioning a more general science of algebra or " universal mathematics ," as a precursor to symbolic logic, that could encompass logical principles and methods symbolically, and mechanize general reasoning.

Although Wittgenstein did not use the term himself, his metaphysical view throughout the Tractatus is commonly referred to as logical atomism.

Although the Vienna Circle's logical positivists appreciated the Tractatus, they argued that the last few passages, including Proposition 7, are confused.

According to the Internet Encyclopedia of Philosophy, Solovyov thought, “ Although empiricism and rationalism (= idealism ) rest on false principles, their respective objective contents, external experience, qua the foundation of natural science, and logical thought, qua the foundation of pure philosophy, are to be synthesized or encompassed along with mystical knowledge in ' integral knowledge ,' what Solovyov terms ' theosophy.

Although originally designed as an indexing and retrieval system, due to its logical structure and scalability, UDC has become one of the most widely used knowledge organization systems in libraries, where it is used for either shelf arrangement, content indexing or both.

Although Bryan had decided on a strategy to gain the nomination — to give a speech which would make him the logical candidate in the eyes of delegates — he faced obstacles along the way.

Although the metric system has changed and developed since its inception, its basic concepts have hardly changed-it was designed for transnational use ; it consisted of a basic set of units of measurement, now known as base units ; derived units were built up from the base units using logical rather than empirical relationships while multiples and submultiples of both base and derived units were decimal based and identified by a standard set of prefixes.

" Although the psychological or epistemological specifics given by Mill through which we build our logical apparatus may not be completely warranted, his explanation still nonetheless manages to demonstrate that there is no way around Kant's a priori logic.

Although federally imposed conditions of reconstruction retrospectively seem logical, there was a widespread belief that southern Unionism would return the seceded states to the Union after the South's military power was broken.

Although some citizens believed Bowling Green was no longer the most practical or logical site for a new courthouse, the commissioners quickly decided to build the third courthouse on the site of the previous one in Bowling Green.

Although the stories were completely independent and shared no common characters they had some similarities that made it logical to combine them.

Although the concepts of a law or principle in nature is borderline to philosophy, and presents the depth to which mathematics can describe nature, scientific laws are considered from a scientific perspective and follow the scientific method ; they " serve their purpose " rather than " questioning reality " ( philosophical ) or " statements of logical absolution " ( mathematical ).

Although this may seem an odd layout, it is in fact logical.

Although he started with a " gut feeling " that Richard was innocent of murdering his nephews, the Princes in the Tower, Fields claims to have investigated the facts as he would for a client he was representing, and he structured the book like a lawyer's brief, identifying the evidence and then drawing the logical implications from the facts.

Although the Duke lost his political influence in the latter stages of the war he still possessed vast prestige abroad, yet his failure to communicate his innermost convictions to his allies or political masters means he must bear some responsibility for the continuance of the war beyond its logical conclusion.

Although this uses only two ( instead of three ) operands per instruction, correspondingly more instructions are then needed to effect various logical operations.

Although it has subsequently grown into another mainstream HOL variant, its logical foundations remain unusually simple.

Although at the time C. S. Lewis wrote Miracles, quantum mechanics ( and physical indeterminism ) was only in the initial stages of acceptance, he stated the logical possibility that if the physical world was proved to be indeterministic this would provide an entry ( interaction ) point into the traditionally viewed closed system, where a scientifically described physically probable / improbable event could be philosophically described as an action of a non physical entity on physical reality.

2. Although normally associated with the crime genre, the term " mystery fiction " may in certain situations refer to a completely different genre, where the focus is on supernatural or thriller mystery ( the solution doesn't have to be logical, and even no crime is involved ).

Although constrained by land mass and topology, the amount of arable land, both regionally and globally, fluctuates due to human and climatic factors such as irrigation, deforestation, desertification, terracing, landfill, and urban sprawl.

Although this topology places a huge overhead on the central hub, with adequate capacity, the hub can handle very high utilization by one device without affecting others.

Although they also have a toroidal magnetic field topology, stellarators are distinct from tokamaks in that they are not azimuthally symmetric.

Although there is no absolute distinction between different areas of topology, the focus here is on general topology.

Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible.

::# Although most networks that are based upon the physical star topology are commonly implemented using a special device such as a hub or switch as the central node ( i. e., the ' hub ' of the star ), it is also possible to implement a network that is based upon the physical star topology using a computer or even a simple common connection point as the ' hub ' or central node.

Although not part of this definition, many authors require that the topology on G be Hausdorff ; this corresponds to the identity map being a closed inclusion ( hence also a cofibration ).

Although such experiments are relatively easy to perform, their mathematical interpretation is far from simple: there may be more than one locally minimizing surface, and they may have non-trivial topology.

Although cohomology is fundamental to modern algebraic topology, its importance was not seen for some 40 years after the development of homology.

Although local homeomorphisms are not as strong as covering maps in this respect, they have many important applications in differential topology ; namely one uses the notion of a local homeomorphism to define a differentiable manifold.

Although Hurewicz knew intimately the topology that was being studied in Poland he chose to go to Vienna to continue his studies.

Although Polish spaces are metrizable, they are not in and of themselves metric spaces ; each Polish space admits many complete metrics giving rise to the same topology, but no one of these is singled out or distinguished.

Although TQFTs were invented by physicists, they are also of mathematical interest, being related to, among other things, knot theory and the theory of four-manifolds in algebraic topology, and to the theory of moduli spaces in algebraic geometry.

Although it uses a different topology and an operational amplifier configured to provide non-unity-gain, it can be analyzed using similar methods as with the generic Sallen – Key topology.

Although the range of equalization functions is governed by the theory of linear filters, the adjustment of those functions and the flexibility with which they can be adjusted varies according to the topology of the circuitry and controls presented to the user.

Although the notion of a point is generally considered fundamental in mainstream geometry and topology, there are some systems that forgo it, e. g. noncommutative geometry and pointless topology.

Although such a shift is symmetry allowed, the Mobius topology required in the transition state prohibits such a shift because it is geometrically impossible, which accounts for the fact that enols do not isomerize without an acid or base catalyst.