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Summary One does not have to look anywhere in order to come to know that 2+3=5. One merely has to think.  Considerations like this underlie the strong intuition that mathematical truths are apriori. That is, very roughly, that our canonical justification or knowledge of them does not essentially rest on experience. Although this claim may seem intuitive, empiricists such as Quine deny it, holding that there is no fundamental epistemological difference between mathematical and non-mathematical knowledge. The question also arises how apriori knowledge and justification in mathematics is possible at all. What exactly do the belief-forming and warrant-generating processes look like? Finally, the rise of computer proofs generates new interesting questions as to the epistemological status of relevant mathematical propositions.
Key works A classical defense of the thesis that arithmetic is apriori is Frege 1953. A sustained defense of a modern philosophy of mathematics combining Platonism with the claim that mathematics is apriori can be found in Wright & Hale 2001. As to the mentioned empiricist picture, a classical paper is Quine 1951. Jenkins 2008 defends the thesis that a version of empiricism can be combined with both mathematical realism and the claim that arithmetic is apriori.
Introductions For more information on the notion of apriority, consult the relevant category on philpapers. For a discussion of different notions of apriority in the philosophy of mathematics, consult e.g. Field 2005. A sustained discussion of the orthodoxy that mathematics is apriori, and the problems it raises in the context of other orthodoxies can be found in Jenkins 2008.
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  1. Mathematics - an Imagined Tool for Rational Cognition.Boris Culina - manuscript
    Analysing several characteristic mathematical models: natural and real numbers, Euclidean geometry, group theory, and set theory, I argue that a mathematical model in its final form is a junction of a set of axioms and an internal partial interpretation of the corresponding language. It follows from the analysis that (i) mathematical objects do not exist in the external world: they are our internally imagined objects, some of which, at least approximately, we can realize or represent; (ii) mathematical truths are not (...)
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  2. The Fundamental Cognitive Approaches of Mathematics.Salvador Daniel Escobedo Casillas - manuscript
    We propose a way to explain the diversification of branches of mathematics, distinguishing the different approaches by which mathematical objects can be studied. In our philosophy of mathematics, there is a base object, which is the abstract multiplicity that comes from our empirical experience. However, due to our human condition, the analysis of such multiplicity is covered by other empirical cognitive attitudes (approaches), diversifying the ways in which it can be conceived, and consequently giving rise to different mathematical disciplines. This (...)
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  3. A Two-Dimensionalist Solution to the Access Problem.Hasen Khudairi - manuscript
    I argue that the two-dimensional intensions of epistemic two-dimensional semantics provide a compelling solution to the access problem.
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  4. The Origin of Europe and the Esprit de Geometrie.Francesco Tampoia - manuscript
    In searching for the origin of Europe and the cultural region/continent that we call “Europe”, at first glance we have to consider at least a double view: on the one hand the geographical understanding which indicates a region or a continent. On the other a certain form of identity and culture described and defined as European. Rodolphe Gasché taking hint from Husserl’s passage ‘Europe is not to be construed simply as a geographical and political entity’ states that a rigorous engagement (...)
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  5. Mary Shepherd on the Role of Proofs in Our Knowledge of First Principles.M. Folescu - forthcoming - Noûs.
    This paper examines the role of reason in Shepherd's account of acquiring knowledge of the external world via first principles. Reason is important, but does not have a foundational role. Certain principles enable us to draw the required inferences for acquiring knowledge of the external world. These principles are basic, foundational and, more importantly, self‐evident and thus justified in other ways than by demonstration. Justificatory demonstrations of these principles are neither required, nor possible. By drawing on textual and contextual evidence, (...)
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  6. On What Ground Do Thin Objects Exist? In Search of the Cognitive Foundation of Number Concepts.Markus Pantsar - forthcoming - Theoria.
  7. Does Anti-Exceptionalism About Logic Entail That Logic is A Posteriori?Stephen Biggs & Jessica Wilson - 2022 - Synthese 1 (1):1--17.
    The debate between exceptionalists and anti-exceptionalists about logic is often framed as concerning whether the justification of logical theories is a priori or a posteriori (for short: whether logic is a priori or a posteriori). As we substantiate (S1), this framing more deeply encodes the usual anti-exceptionalist thesis that logical theories, like scientific theories, are abductively justified, coupled with the common supposition that abduction is an a posteriori mode of inference, in the sense that the epistemic value of abduction is (...)
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  8. Introduction to Knowledge, Number and Reality. Encounters with the Work of Keith Hossack.Nils Kürbis, Jonathan Nassim & Bahram Assadian - 2022 - In Nils Kürbis, Jonathan Nassim & Bahram Assadian (eds.), Knowledge, Number and Reality. Encounters with the Work of Keith Hossack. London: Bloomsbury Academic. pp. 1-30.
    The Introduction to "Knowledge, Number and Reality. Encounters with the Work of Keith Hossack" provides an overview over Hossack's work and the contributions to the volume.
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  9. Color and a Priori Knowledge.Brian Cutter - 2021 - Philosophical Studies 178 (1):293-315.
    Some truths about color are knowable a priori. For example, it is knowable a priori that redness is not identical to the property of being square. This extremely modest and plausible claim has significant philosophical implications, or so I shall argue. First, I show that this claim entails the falsity of standard forms of color functionalism, the view that our color concepts are functional concepts that pick out their referents by way of functional descriptions that make reference to the subjective (...)
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  10. Groundwork for a Fallibilist Account of Mathematics.Silvia De Toffoli - 2021 - Philosophical Quarterly 7 (4):823-844.
    According to the received view, genuine mathematical justification derives from proofs. In this article, I challenge this view. First, I sketch a notion of proof that cannot be reduced to deduction from the axioms but rather is tailored to human agents. Secondly, I identify a tension between the received view and mathematical practice. In some cases, cognitively diligent, well-functioning mathematicians go wrong. In these cases, it is plausible to think that proof sets the bar for justification too high. I then (...)
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  11. Neues System der philosophischen Wissenschaften im Grundriss. Band II: Mathematik und Naturwissenschaft.Dirk Hartmann - 2021 - Paderborn: Mentis.
    Volume II deals with philosophy of mathematics and general philosophy of science. In discussing theoretical entities, the notion of antirealism formulated in Volume I is further elaborated: Contrary to what is usually attributed to antirealism or idealism, the author does not claim that theoretical entities do not really exist, but rather that their existence is not independent of the possibility to know about them.
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  12. Apriority and Essential Truth.Tristan Grøtvedt Haze - 2020 - Metaphysica 21 (1):1-8.
    There is a line of thought, neglected in recent philosophy, according to which a priori knowable truths such as those of logic and mathematics have their special epistemic status in virtue of a certain tight connection between their meaning and their truth. Historical associations notwithstanding, this view does not mandate any kind of problematic deflationism about meaning, modality or essence. On the contrary, we should be upfront about it being a highly debatable metaphysical idea, while nonetheless insisting that it be (...)
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  13. Metamathematics and the Philosophical Tradition, by William Boos, ed. Florence S. Boos. Berlin: De Gruyter. 2018. [REVIEW]Lydia Patton - 2020 - Philosophia 48 (4):1-4.
    William Boos (1943–2014) was a mathematician, set theorist, and philosopher. His work is at the intersection of these fields. In particular, Boos looks at the classic problems of epistemology through the lens of the axiomatic method in mathematics and physics, or something resembling that method.
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  14. Kitcher, Mathematics, and Apriority.Jeffrey Roland - 2019 - Erkenntnis 84 (3):687-702.
    Philip Kitcher has argued against the apriority of mathematical knowledge in a number of places. His arguments rely on a conception of mathematical knowledge as embedded in a historical tradition and the claim that this sort of embedding compromises apriority. In this paper, I argue that tradition dependence of mathematical knowledge does not compromise its apriority. I further identify the factors which appear to lead Kitcher to argue as he does.
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  15. What is the Benacerraf Problem?Justin Clarke-Doane - 2017 - In Fabrice Pataut (ed.), New Perspectives on the Philosophy of Paul Benacerraf: Truth, Objects, Infinity. Springer Verlag.
    In "Mathematical Truth", Paul Benacerraf articulated an epistemological problem for mathematical realism. His formulation of the problem relied on a causal theory of knowledge which is now widely rejected. But it is generally agreed that Benacerraf was onto a genuine problem for mathematical realism nevertheless. Hartry Field describes it as the problem of explaining the reliability of our mathematical beliefs, realistically construed. In this paper, I argue that the Benacerraf Problem cannot be made out. There simply is no intelligible problem (...)
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  16. Our Incorrigible Ontological Relations and Categories of Being.Julian M. Galvez Bunge (ed.) - 2017 - USA: Amazon.
    The purpose of this book is to address the controversial issues of whether we have a fixed set of ontological categories and if they have some epistemic value at all. Which are our ontological categories? What determines them? Do they play a role in cognition? If so, which? What do they force to presuppose regarding our world-view? If they constitute a limit to possible knowledge, up to what point is science possible? Does their study make of philosophy a science? Departing (...)
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  17. Knowledge of Abstract Objects in Physics and Mathematics.Michael Shaffer - 2017 - Acta Analytica 32 (4):397-409.
    In this paper a parallel is drawn between the problem of epistemic access to abstract objects in mathematics and the problem of epistemic access to idealized systems in the physical sciences. On this basis it is argued that some recent and more traditional approaches to solving these problems are problematic.
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  18. A Framework for Implicit Definitions and the A Priori.Philip A. Ebert - 2016 - In Philip A. Ebert & Marcus Rossberg (eds.), Abstractionism. Oxford: Oxford University Press. pp. 133--160.
  19. Introduction to Abstractionism.Philip A. Ebert & Marcus Rossberg - 2016 - In Philip A. Ebert & Marcus Rossberg (eds.), Abstractionism. Oxford: Oxford University Press. pp. 3-33.
  20. The Modal Status of Contextually A Priori Arithmetical Truths.Markus Pantsar - 2016 - In Andrea Sereni & Francesca Boccuni (eds.), Objectivity, Realism, and Proof. Springer International Publishing. pp. 67-79.
    In Pantsar (2014), an outline for an empirically feasible epistemological theory of arithmetic is presented. According to that theory, arithmetical knowledge is based on biological primitives but in the resulting empirical context develops an essentially a priori character. Such contextual a priori theory of arithmetical knowledge can explain two of the three characteristics that are usually associated with mathematical knowledge: that it appears to be a priori and objective. In this paper it is argued that it can also explain the (...)
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  21. The Modal Status of Contextually A Priori Arithmetical Truths.Markus Pantsar - 2016 - In Francesca Boccuni & Andrea Sereni (eds.), Objectivity, Realism, and Proof. Filmat Studies in the Philosophy of Mathematics. Springer Verlag.
    In Pantsar, an outline for an empirically feasible epistemological theory of arithmetic is presented. According to that theory, arithmetical knowledge is based on biological primitives but in the resulting empirical context develops an essentially a priori character. Such contextual a priori theory of arithmetical knowledge can explain two of the three characteristics that are usually associated with mathematical knowledge: that it appears to be a priori and objective. In this paper it is argued that it can also explain the third (...)
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  22. Against Mathematical Convenientism.Seungbae Park - 2016 - Axiomathes 26 (2):115-122.
    Indispensablists argue that when our belief system conflicts with our experiences, we can negate a mathematical belief but we do not because if we do, we would have to make an excessive revision of our belief system. Thus, we retain a mathematical belief not because we have good evidence for it but because it is convenient to do so. I call this view ‘ mathematical convenientism.’ I argue that mathematical convenientism commits the consequential fallacy and that it demolishes the Quine-Putnam (...)
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  23. Review of The Art of the Infinite by R. Kaplan, E. Kaplan 324p(2003).Michael Starks - 2016 - In Suicidal Utopian Delusions in the 21st Century: Philosophy, Human Nature and the Collapse of Civilization-- Articles and Reviews 2006-2017 2nd Edition Feb 2018. Michael Starks. pp. 619.
    This book tries to present math to the millions and does a pretty good job. It is simple and sometimes witty but often the literary allusions intrude and the text bogs down in pages of relentless math--lovely if you like it and horrid if you don´t. If you already know alot of math you will still probably find the discussions of general math, geometry, projective geometry, and infinite series to be a nice refresher. If you don´t know any and don´t (...)
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  24. Closure of A Priori Knowability Under A Priori Knowable Material Implication.Jan Heylen - 2015 - Erkenntnis 80 (2):359-380.
    The topic of this article is the closure of a priori knowability under a priori knowable material implication: if a material conditional is a priori knowable and if the antecedent is a priori knowable, then the consequent is a priori knowable as well. This principle is arguably correct under certain conditions, but there is at least one counterexample when completely unrestricted. To deal with this, Anderson proposes to restrict the closure principle to necessary truths and Horsten suggests to restrict it (...)
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  25. The Eleatic and the Indispensabilist.Russell Marcus - 2015 - Theoria : An International Journal for Theory, History and Fundations of Science 30 (3):415-429.
    The debate over whether we should believe that mathematical objects exist quickly leads to the question of how to determine what we should believe. Indispensabilists claim that we should believe in the existence of mathematical objects because of their ineliminable roles in scientific theory. Eleatics argue that only objects with causal properties exist. Mark Colyvan’s recent defenses of Quine’s indispensability argument against some contemporary eleatics attempt to provide reasons to favor the indispensabilist’s criterion. I show that Colyvan’s argument is not (...)
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  26. Review of Space, Time, and Number in the Brain. [REVIEW]Carlos Montemayor & Rasmus Grønfeldt Winther - 2015 - Mathematical Intelligencer 37 (2):93-98.
    Albert Einstein once made the following remark about "the world of our sense experiences": "the fact that it is comprehensible is a miracle." (1936, p. 351) A few decades later, another physicist, Eugene Wigner, wondered about the unreasonable effectiveness of mathematics in the natural sciences, concluding his classic article thus: "the miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve" (1960, p. 14). (...)
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  27. Lakatos’ Quasi-Empiricism in the Philosophy of Mathematics.Michael J. Shaffer - 2015 - Polish Journal of Philosophy 9 (2):71-80.
    Imre Lakatos' views on the philosophy of mathematics are important and they have often been underappreciated. The most obvious lacuna in this respect is the lack of detailed discussion and analysis of his 1976a paper and its implications for the methodology of mathematics, particularly its implications with respect to argumentation and the matter of how truths are established in mathematics. The most important themes that run through his work on the philosophy of mathematics and which culminate in the 1976a paper (...)
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  28. Does The Necessity of Mathematical Truths Imply Their Apriority?Mark McEvoy - 2013 - Pacific Philosophical Quarterly 94 (4):431-445.
    It is sometimes argued that mathematical knowledge must be a priori, since mathematical truths are necessary, and experience tells us only what is true, not what must be true. This argument can be undermined either by showing that experience can yield knowledge of the necessity of some truths, or by arguing that mathematical theorems are contingent. Recent work by Albert Casullo and Timothy Williamson argues (or can be used to argue) the first of these lines; W. V. Quine and Hartry (...)
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  29. The Defeater Version of Benacerraf’s Problem for a Priori Knowledge.Joshua C. Thurow - 2013 - Synthese 190 (9):1587-1603.
    Paul Benacerraf’s argument that mathematical realism is apparently incompatible with mathematical knowledge has been widely thought to also show that a priori knowledge in general is problematic. Although many philosophers have rejected Benacerraf’s argument because it assumes a causal theory of knowledge, some maintain that Benacerraf nevertheless put his finger on a genuine problem, even though he didn’t state the problem in its most challenging form. After diagnosing what went wrong with Benacerraf’s argument, I argue that a new, more challenging, (...)
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  30. Kant's Views on Non-Euclidean Geometry.Michael Cuffaro - 2012 - Proceedings of the Canadian Society for History and Philosophy of Mathematics 25:42-54.
    Kant's arguments for the synthetic a priori status of geometry are generally taken to have been refuted by the development of non-Euclidean geometries. Recently, however, some philosophers have argued that, on the contrary, the development of non-Euclidean geometry has confirmed Kant's views, for since a demonstration of the consistency of non-Euclidean geometry depends on a demonstration of its equi-consistency with Euclidean geometry, one need only show that the axioms of Euclidean geometry have 'intuitive content' in order to show that both (...)
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  31. Scientific Heritage: The Reception and Transmission of Euclidian Geometry in Western Civilization.Zbigniew Król - 2012 - Dialogue and Universalism 22 (4):41-65.
    This paper presents sources pertinent to the transmission of Euclid’s Elements in Western medieval civilization. Some important observations follow from the pure description of the sources concerning the development of mathematics, e.g., the text of the Elements was supplemented with new axioms, proofs and theorems as if an “a priori skeleton” lost in Dark Ages was reconstructed and rediscovered during the late Middle Ages. Such historical facts indicate the aprioricity of mathematics.
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  32. Friedman on Implicit Definition: In Search of the Hilbertian Heritage in Philosophy of Science.Woosuk Park - 2012 - Erkenntnis 76 (3):427-442.
    Michael Friedman’s project both historically and systematically testifies to the importance of the relativized a priori. The importance of implicit definitions clearly emerges from Schlick’s General Theory of Knowledge . The main aim of this paper is to show the relationship between both and the relativized a priori through a detailed discussion of Friedman’s work. Succeeding with this will amount to a contribution to recent scholarship showing the importance of Hilbert for Logical Empiricism.
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  33. Computer, Proof, and Testimony.Kai-Yee Wong - 2012 - Studies in Logic 5 (1):50-67.
    It has been claimed that computer-assisted proof utilizes empirical evidence in a manner unheard of in traditional mathematics and therefore its employment forces us to modify our conception of proof. This paper provides a critical survey of some arguments for this claim. It starts by revisiting a well known paper by Thomas Tymoczko on the computer proof of the Four-Color Theorem. Drawing on some ideas from the works of Tyler Burge and others, it then considers a way to see the (...)
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  34. Justified Concepts and the Limits of the Conceptual Approach to the A Priori.Darren Bradley - 2011 - Croatian Journal of Philosophy 11 (3):267-274.
    Carrie Jenkins (2005, 2008) has developed a theory of the a priori that she claims solves the problem of how justification regarding our concepts can give us justification regarding the world. She claims that concepts themselves can be justified, and that beliefs formed by examining such concepts can be justified a priori. I object that we can have a priori justified beliefs with unjustified concepts if those beliefs have no existential import. I then argue that only beliefs without existential import (...)
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  35. Grounding Concepts: An Empirical Basis for Arithmetical Knowledge – C.S. Jenkins.James Robert Brown & James Davies - 2011 - Philosophical Quarterly 61 (242):208-211.
  36. Bolzano Versus Kant: Mathematics as a Scientia Universalis.Paola Cantù - 2011 - Philosophical Papers Dedicated to Kevin Mulligan.
    The paper discusses some changes in Bolzano's definition of mathematics attested in several quotations from the Beyträge, Wissenschaftslehre and Grössenlehre: is mathematics a theory of forms or a theory of quantities? Several issues that are maintained throughout Bolzano's works are distinguished from others that were accepted in the Beyträge and abandoned in the Grössenlehre. Changes are interpreted as a consequence of the new logical theory of truth introduced in the Wissenschaftslehre, but also as a consequence of the overcome of Kant's (...)
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  37. Grounding Concepts: The Problem of Composition.Gábor Forrai - 2011 - Philosophia 39 (4):721-731.
    In a recent book C.S. Jenkins proposes a theory of arithmetical knowledge which reconciles realism about arithmetic with the a priori character of our knowledge of it. Her basic idea is that arithmetical concepts are grounded in experience and it is through experience that they are connected to reality. I argue that the account fails because Jenkins’s central concept, the concept for grounding, is inadequate. Grounding as she defines it does not suffice for realism, and by revising the definition we (...)
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  38. A Priori.Edwin Mares - 2011 - Routledge.
    In recent years many influential philosophers have advocated that philosophy is an a priori science. Yet very few epistemology textbooks discuss a priori knowledge at any length, focusing instead on empirical knowledge and empirical justification. As a priori knowledge has moved centre stage, the literature remains either too technical or too out of date to make up a reasonable component of an undergraduate course. Edwin Mares book aims to rectify this. This book seeks to make accessible to students the standard (...)
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  39. Russell on Logicism and Coherence.Conor Mayo-Wilson - 2011 - Russell: The Journal of Bertrand Russell Studies 31 (1):89-106.
    According to Quine, Charles Parsons, Mark Steiner, and others, Russell's logicist project is important because, if successful, it would show that mathematical theorems possess desirable epistemic properties often attributed to logical theorems, such as a prioricity, necessity, and certainty. Unfortunately, Russell never attributed such importance to logicism, and such a thesis contradicts Russell's explicitly stated views on the relationship between logic and mathematics. This raises the question: what did Russell understand to be the philosophical importance of logicism? Building on recent (...)
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  40. Review of Discourse on a New Method: Reinvigorating the Marriage of History and Philosophy of Science. [REVIEW]Lydia Patton - 2011 - Notre Dame Philosophical Reviews.
    That the history and the philosophy of science have been united in a form of disciplinary marriage is a fact. There are pressing questions about the state of this union. Discourse on a New Method: Reinvigorating the Marriage of History and Philosophy of Science is a state of the union address, but also an articulation of compelling and well-defended positions on strategies for making progress in the history and philosophy of science.
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  41. The Paradox of Infinite Given Magnitude: Why Kantian Epistemology Needs Metaphysical Space.Lydia Patton - 2011 - Kant-Studien 102 (3):273-289.
    Kant's account of space as an infinite given magnitude in the Critique of Pure Reason is paradoxical, since infinite magnitudes go beyond the limits of possible experience. Michael Friedman's and Charles Parsons's accounts make sense of geometrical construction, but I argue that they do not resolve the paradox. I argue that metaphysical space is based on the ability of the subject to generate distinctly oriented spatial magnitudes of invariant scalar quantity through translation or rotation. The set of determinately oriented, constructed (...)
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  42. C. S. Jenkins, Grounding Concepts: An Empirical Basis for Arithmetical Knowledge Reviewed By.Manuel Bremer - 2010 - Philosophy in Review 30 (3):205-207.
  43. Grassmann’s Epistemology: Multiplication and Constructivism.Paola Cantù - 2010 - In Hans-Joachim Petsche (ed.), From Past to Future: Graßmann's Work in Context.
    The paper aims to establish if Grassmann’s notion of an extensive form involved an epistemological change in the understanding of geometry and of mathematical knowledge. Firstly, it will examine if an ontological shift in geometry is determined by the vectorial representation of extended magnitudes. Giving up homogeneity, and considering geometry as an application of extension theory, Grassmann developed a different notion of a geometrical object, based on abstract constraints concerning the construction of forms rather than on the homogeneity conditions required (...)
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  44. Grounding Concepts, by C. S. Jenkins.: Book Reviews. [REVIEW]Albert Casullo - 2010 - Mind 119 (475):805-810.
    (No abstract is available for this citation).
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  45. Descriptions and Unknowability.Jan Heylen - 2010 - Analysis 70 (1):50-52.
    In a recent paper Horsten embarked on a journey along the limits of the domain of the unknowable. Rather than knowability simpliciter, he considered a priori knowability, and by the latter he meant absolute provability, i.e. provability that is not relativized to a formal system. He presented an argument for the conclusion that it is not absolutely provable that there is a natural number of which it is true but absolutely unprovable that it has a certain property. The argument depends (...)
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  46. Review of Grounding Concepts by C. S. Jenkins. [REVIEW]Joshua Schechter - 2010 - Notre Dame Philosophical Reviews 2010 (5).
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  47. Review of C. S. Jenkins, Grounding Concepts: An Empirical Basis for Arithmetical Knowledge[REVIEW]Neil Tennant - 2010 - Philosophia Mathematica 18 (3):360-367.
    This book is written so as to be ‘accessible to philosophers without a mathematical background’. The reviewer can assure the reader that this aim is achieved, even if only by focusing throughout on just one example of an arithmetical truth, namely ‘7+5=12’. This example’s familiarity will be reassuring; but its loneliness in this regard will not. Quantified propositions — even propositions of Goldbach type — are below the author’s radar.The author offers ‘a new kind of arithmetical epistemology’, one which ‘respects (...)
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  48. Carrie Jenkins, Grounding Concepts. An Empirical Basis for Arithmetical Knowledge. Oxford, Oxford University Press, 2008.Lorenz Demey - 2009 - Tijdschrift Voor Filosofie 71 (4):796-798.
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  49. Synthetic a priori judgments.Alessandro Giordani - 2009 - Rivista di Filosofia Neo-Scolastica 17:297 - 313.
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  50. Benacerraf’s Dilemma and Informal Mathematics.Gregory Lavers - 2009 - Review of Symbolic Logic 2 (4):769-785.
    This paper puts forward and defends an account of mathematical truth, and in particular an account of the truth of mathematical axioms. The proposal attempts to be completely nonrevisionist. In this connection, it seeks to satisfy simultaneously both horns of Benacerrafs work on informal rigour. Kreisel defends the view that axioms are arrived at by a rigorous examination of our informal notions, as opposed to being stipulated or arrived at by trial and error. This view is then supplemented by a (...)
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