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This is a list of lowercase non-hyphenated single words, lacking English entries in the English Wiktionary as of the most recent database dump, found in the 2008-05 issue of Erkenntnis.
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44582 tokens ‧ 37823 valid lowercase tokens ‧ 4235 types ‧ 77 (~ 1.818%) words before cleaning ‧
2008, Bernd Buldt, Benedikt Löwe and Thomas Müller, “Towards a New Epistemology of Mathematics”, in Erkenntnis, volume 68, number 3, DOI:10.1007/s10670-008-9101-6:
- According to the traditional philosophical analysis, mathematical theorems are a priori truths about acausal, non-spatio-temporal objects.
2008, Kajsa Bråting and Johanna Pejlare, “Visualizations in Mathematics”, in Erkenntnis, volume 68, number 3, DOI:10.1007/s10670-008-9104-3:
- He considered the function strip as a collection of functions that are approximatively equal to each other and he considered a function to represent a strip if all its values belongs to the strip.
- The first historical case study presented pertains to the criticism and the eventual decline of visualization in mathematics triggered by geometrical and analytical argumentations drifting apart in the course of the rigourization of analysis in the 19th century.
- In his “The role of axioms in mathematics”, Kenny Easwaran wants to correct another aspect that received wisdom has to offer on axiomatics.
2008, Dirk Schlimm, “On Abstraction and the Importance of Asking the Right Research Questions: Could Jordan have Proved the Jordan-Hölder Theorem?”, in Erkenntnis, volume 68, number 3, DOI:10.1007/s10670-008-9108-z:
- The subsequent spread of the abstract notion involved the transfer of particular theorems for substitution groups to the abstract case, the acknowledgment of the notion of group as a fundamental structure in algebra, and detailed investigations of different axiomatizations.
- The ideal of uncontroversial checkability of mathematical arguments, however, seems to be related to formal derivations rather than scribblings on napkins.
2008, Bart Van Kerkhove and Jean Paul Van Bendegem, “Pi on Earth, or Mathematics in the Real World”, in Erkenntnis, volume 68, number 3, DOI:10.1007/s10670-008-9102-5:
- 11 Next to this quest for a formal proof however, since about the middle of the eighteenth century, substantial efforts have been done to check an increasing amount of specific numbers upon this particular way of decomposability.
- Although, in mathematics, one is not supposed to settle for anything short of absolute proof, even staunch deductivists might appreciate the potential value of this way of proceeding.
2008, Brendan Larvor, “What can the Philosophy of Mathematics Learn from the History of Mathematics?”, in Erkenntnis, volume 68, number 3, DOI:10.1007/s10670-008-9107-0:
- They think they are doing a thing honour when they dehistoricize it, sub specie aeterni —when they make a mummy of it.
- Topics that have been found to become philosophically informing in this respect include digitalization, complexity, feasibility, and induction.
2008, Kenny Easwaran, “The Role of Axioms in Mathematics”, in Erkenntnis, volume 68, number 3, DOI:10.1007/s10670-008-9106-1:
- With foundational axioms, realists about abstract numbers can share the theorems of Peano arithmetic with fictionalists, while with structural axioms, group theorists can share theorems about topological spaces with set theorists.
- In Wilhelmus ( 2007 ), the author investigated the philosophical question “Is formalizability of an argument a necessary condition for mathematical knowledge?
- A radical denial of foundationalism is offered by social constructivism (Ernest 1998 ), an approach that many researchers in mathematics education embrace.
- These inner-scientific episodes, as witnessed by their public reflection, suggest that one should consider a revision of the foundationalist epistemology of mathematics.
- Turning to the case of mathematics, even if there are mathematical objects or truths laid up in a Platonic heaven, or gapless proofs that an ideal mathematician could give ‘in principle’, such ideal items are no more historically effective than the contents of scripture.
- The historian of philosophy (in Williams’ sense) uses historiographic techniques to place texts and actors in their proper contexts and understand them in their proper times, but only insofar as this serves the philosophical goals of the enquiry.
- The informal dimension inherent to these fields of inquiry is captured by notorious notions such as unpredictability, nonlinearity, chaos, emergence, or incomputability.
- However, whereas Van Bendegem and Van Kerkhove conclude by saying that experimental methods and an empirical basis form an indispensible backdrop for mathematical practice, Baker goes the other way and says that the fact that mathematicians use experiments in the context of discovery is “compatible with the view that mathematics is a priori and deductive at its core”.
- Their solution was to adopt Russell’s logicist thesis that every mathematical concept can be defined in the language of logic and that every mathematical proof can be replaced by a purely logical derivation using the logical definitions of the concepts involved; hence the label “ Logical Empiricism”.
- The ‘intuitively evident’ fact that a function that varies continuously must be piecewise monotonic was used, and differentiability and monotonicity were linked together.
- Most likely we will wind up abandoning the task of keeping track of price altogether and complete the metamorphosis to nonrigorous mathematics.
- Because it involves the addition of numbers defined in terms of multiplication, this problem is as hard to tackle as it is easy to phrase and grasp (as for the latter, it only takes one to master the concepts of evenness and primehood).
- The prover belongs to a family of checking devices, Turing machines or sequences of these, that are capable of establishing the probable correctness of solutions for very large classes of problems.
- In the second scenario, they employ Malament–Hogarth spacetimes, a theory that recently gained a lot of attention in the computability community as these solutions to the Einstein equations of General Relativity allowing for an infinite amount of time to pass in what is a finite amount of time for an observer.
- One might object that this scenario is truly a fiction, but the recent literature on supertasks shows that, although their reality is definitely still not established, at least its plausibility can be argued for.
- The fourth topic, induction, touches upon the alleged experimental and other non-deductive dimensions to mathematical activity, involving such techniques as the brute manipulation of numbers, probability arguments, or visual proofs, and putting to the test the epistemic principles of surveyability and understanding.
- For example, this criticism also applies to deductive proofs so long or complex that they are not globally surveyable by any one individual, however intellectually supreme.
- We began with history and philosophy as opposite poles (one temporal and particular, the other tenseless and universal).
2008, Alan Baker, “Experimental Mathematics”, in Erkenntnis, volume 68, number 3, DOI:10.1007/s10670-008-9109-y:
- If explanation is indeed tied closely to unification then it is not hard to see how the disjunctiveness characteristic of computer proof tends to yield proofs that are also considered relatively unexplanatory by mathematicians.
- The most common reason for resorting to computers to help with proving a given result is because we are faced with an unfeasibly large number of particular cases that need to be worked through in order to verify a general claim.
- More serious, perhaps, for the computational characterization is that many uses of computation fall unproblematically under traditional mathematical methodology and seem to have nothing to do with experimental mathematics.
- Kant was concerned about certain metaphysical notions, but we think that usurpatory concepts reflect a general phenomenon of language as it evolves over time.
- persistance - typo only?
- We can only aim at part of the answer here, contending that, at the very least, this persistance has to do somehow with the hope of increasing insight in problems at hand, more particularly by exploring the mathematical realm surrounding them.