This ICME Topical Survey is designed to provide an overview of contemporary research in the philosophy of mathematics education. This is a broad cluster of overlapping but at times disparate themes.
In the first instance, this publication exposes some of the problems and questions in mathematics education that the philosophy of mathematics education clarifies, illuminates and sometimes helps to solve. A metaphor for what is offered is a three tier pyramid. Finally, at the base level is the full spread of research and its results, books, journals, papers, conference presentations and other activities that make up the subfield, the philosophy of mathematics education, which is beyond the scope of this publication and conference.
This publication thus briefly sketches some of the topics, problems and areas of active research the apex and through this point to some of what will be offered at the conference the middle level. In doing so it serves as an introduction to the extent of the sub-field overall, through references to current publications and classic literature the base of the pyramid.
Why the philosophy of mathematics education? What does it offer? The philosophy of any activity comprises its aims or rationale. Given our shared commitment to the teaching and learning of mathematics it is vital ask: What is the purpose of teaching and learning mathematics? What do we value in mathematics and its teaching and learning? Why do we engage in these practices and what do we hope will be achieved? The sub-field can also help us uncover whatever implicit assumptions and priorities underlie mathematics education. These can including paradigmatic assumptions of which we may be unaware, but that can be identified through, let us say, a philosophical archaeology.
What is the significance of information and communication technology in the teaching and learning of mathematics? How and to what extent is social justice promulgated by these activities and this field of study? What deep and often unacknowledged assumptions underlie mathematics education research and practice? It enables people to see beyond official stories about the society, mathematics, and education.
In addition to these special interests Brazil has an active research community in the theoretical aspects of the philosophy of mathematics education and their applications to policy and practice. Recent developments in terms of organisation, basis and research orientation of one strand in the area are sketched in Sect. The philosophy of mathematics education can be interpreted both narrowly and more widely.
Understood narrowly the philosophy of some activity is its aim or rationale. So in the narrow sense the philosophy of mathematics education concerns the aims or rationale behind the practice of teaching mathematics. However, the aims, goals, purposes, rationales, etc. Since the teaching of mathematics is a widespread and highly organised social activity, its aims, goals, purposes, rationales, and so on, need to be related to social groups and society in general, while acknowledging that there are multiple and divergent aims and goals among different persons and groups. Aims are expressions of values, and thus the educational and social values of society or some part of it are implicated in this enquiry.
Philosophy of mathematics applied to mathematics education or to education in general. Philosophy of education applied to mathematics education Brown The application of philosophical concepts or methods, such as a critical attitude to claims as well as detailed conceptual analyses of the concepts, theories, methodology or results of mathematics education research, and of mathematics itself Ernest ; Skovsmose Philosophy is about systematic analysis and the critical examination of fundamental problems.
It involves the exercise of the mind and intellect, including thought, enquiry, reasoning and its results: judgements, conclusions, beliefs and knowledge. There are many ways in which such processes as well as the substantive theories, concepts and results of past enquiry can be applied to and within mathematics education. Why does philosophy matter? Why does theory in general matter? First, because it helps to structure research and inquiries in an intelligent and well grounded way, offering a secure basis for knowledge. It provides an overall structure fitting the results of cutting edge research into the hard-won body of accepted knowledge.
But in addition, it enables people to see beyond the official stories about the world, about society, economics, education, mathematics, teaching and learning.
It enables commonly accepted notions to be probed, questioned and implicit assumptions, ideological distortions or unintended prejudices to be revealed and challenged. It also, most importantly, enables us to imagine alternatives. This analysis suggests that the philosophy of mathematics education should attend not only to the aims and purposes of the teaching and learning of mathematics the narrow sense or even just the philosophy of mathematics and its implications for educational practice.
It suggests that we should look more widely for philosophical and theoretical tools for understanding all aspects of the teaching and learning of mathematics and its milieu. The last questions mathematics education itself. Is mathematics education a discipline, a field of enquiry, an interdisciplinary area, a domain of extra-disciplinary applications, or what? Is it a science, social science, art or humanity, or none or all of these? What is its relationship with other disciplines such as philosophy, mathematics, sociology, psychology, linguistics, anthropology, etc.?
How do we come to know in mathematics education? What is the basis for knowledge claims in research in mathematics education? What research methods and methodologies are employed and what is their philosophical basis and status? How does the mathematics education research community judge knowledge claims?
What standards are applied? How do these relate to the standards used in research in general education, social sciences, humanities, arts, mathematics, the physical sciences and applied sciences such as medicine, engineering and technology? What is the role and function of the researcher in mathematics education? What is the status of theories in mathematics education? Which is better? What impact on mathematics education have modern developments in philosophy had, including phenomenology, critical theory, post-structuralism, post-modernism, Hermeneutics, semiotics, linguistic philosophy, etc.?
What is the impact of research in mathematics education on other disciplines? What do adjacent STEM education subjects science, technology, engineering and mathematics education have in common, and how do they differ? Can the philosophy of mathematics education have any impact on the practices of teaching and learning of mathematics, on research in mathematics education, or on other disciplines? What is the status of the philosophy of mathematics education itself?
How central is mathematics to research in mathematics education? Does mathematics education have an adequate and suitable philosophy of technology in order to accommodate the deep issues raised by information and communication technologies? These five questions encompass much of what is important for the philosophy of mathematics education to consider and explore. These sets are not wholly discrete, as various areas of overlap would be revealed.
Many of the sub-questions involved omitted here except for question 5, but see Ernest are not essentially philosophical, in that they can also be addressed and explored in ways that foreground other disciplinary perspectives, such as sociology and psychology. However, when such questions are approached philosophically, they become part of the business of the philosophy of mathematics education. Also, the philosophy of mathematics education is an area where interdisciplinary questions can be addressed and tentative answers explored. Lastly, if there were a move to exclude any of these questions right from the outset without considering them it would risk adopting or promoting a particular philosophical position, a particular ideology or philosophy of mathematics education.
Exclusionary tactics across social and conceptual domains often have an unacknowledged hidden agenda, and mostly do not serve the advance of knowledge. Thus it could consider research and theory in mathematics education according to whether it draws on ontology and metaphysics; epistemology and learning theory; social and political philosophy; aesthetics, ethics and axiology the philosophy of values more generally; the methodology of mathematics education research; or other branches of philosophy.
Ontology and metaphysics have as yet been little applied in mathematics education research Ernest Work drawing on aesthetics is still in its infancy Ernest , ; Sinclair However extensive uses of epistemology and learning theory, social and political philosophy, ethics and methodology can be found in mathematics education research and literature.
The philosophy of applied mathematics | ujomerywar.tk
In addition to the contributions of the substantive branches of philosophy to mathematics education, there are also benefits to be gained from applying philosophical styles of thinking in our research. For example, many of the constructs we utilise need careful conceptual analysis and critique. Understanding is perhaps the most basic and obvious of these terms, so what can deconstructing it add to our knowledge? Can it contain hidden assumptions and pitfalls? In what way does this capture its meaning?
But secondly, there is an ideological assumption that understanding a concept or skill is better, deeper and more valuable than simply being able to use or perform it successfully. However his co-originator of the distinction Mellin-Olsen used it to distinguish the modes of thinking of academic students from that of apprentices, thus bringing in a social context and even a social class dimension to the distinction, and imposing less of an implicit received valuation.
If we want to assert the superiority of relational understanding over instrumental understanding it needs to be done on the basis of a reasoned argument, and not taken for granted as obvious. Given current challenges to the underlying theories of learning, the assumption that relational understanding is superior stands in need of justification.
Understanding is produced as hierarchical, particularly in relation to gender, social class and ability. Pupils and understanding are tied up with notions such as gender, confidence and emotions. Llewellyn , pp. Although what I have offered here is not conclusive, what the example shows is that even a widely presupposed good in the discourse of mathematics education, the concept of understanding, is a worthwhile target of philosophical analysis and critique.
Although such analysis does not mean that we have to abandon the concept, it does mean that we need to be aware of the penumbra of meanings revealed and aporias unleashed through its deconstruction. We need to use the term with caution and precision, clarifying or sidestepping its troubling connotations and implications.
It can clear the conceptual landscape of unnoticed obstacles and perform the hygienic function of targeting, inoculating and neutralizing potentially toxic ideas circulating, like viruses, in our discourse. In Sect. However, this little assay into the topic is just the beginning, for there are many more unanswered questions. For example: what are the overall responsibilities of mathematics education as an overall field of study and practice, and what is the responsibility of our own subfield, the philosophy of mathematics education?
What are the responsibilities of mathematics education researchers? Does this depend on our philosophical stances, whether we see ourselves as critical public intellectuals or as functional academics probing deeper into narrow specialisms? Philosophy emerged from the dialectics of the ancient Greeks where commonplace beliefs and unanalysed concepts were interrogated and scrutinised, where the role of the rulers was questioned and challenged through speaking truth to power.
Thus the role of the philosophy of mathematics education is to analyse, question, challenge, and critique the claims of mathematics education practice, policy and research. Our job is to unearth hidden assumptions and presuppositions, and by making them overt and visible, to enable researchers and practitioners to boldly go beyond their own self-imposed limits, beyond the unquestioned conceptual boundaries installed by the discourse of our field, to work towards realizing their own dreams, visions and ideals. Critical mathematics education works for social justice in whatever form possible, and it addresses mathematics critically in all its appearances and applications.
It has developed in many directions, and as a consequence the very notion of critical mathematics education refers to a broad range of approaches. One can think of mathematics education for social justice Sriraman ; Wager and Stinson ; pedagogy of dialogue and conflict Vithal ; radical mathematics Frankenstein ; responsive mathematics education Greer et al. Explicit formulations of critical mathematics education are found in, for instance, Frankenstein and Skovsmose , a.
Critical mathematics education can be characterised in terms of concerns, and let me mention some related to mathematics , students , teachers , and society. Mathematics can be brought in action in technology, production, automatisation, decision making, management, economic transaction, daily routines, information procession, communication, security procedures, etc.
In fact mathematics in action plays a part in all spheres of life. It is a concern of a critical mathematics education to address mathematics in its very many different forms of applications and practices. There are no qualities—like, for instance, objectivity and neutrality—that automatically can be associated to mathematics. Mathematics-based actions can have all kind of qualities, being risky, reliable, dangerous, suspicious, misleading, expensive, brutal, benevolent, profit-generating, etc.
Mathematics-based action can serve any kind of interest. As with any form of action, so also mathematics in action is in need of being carefully criticized. This applies to any form of mathematics: everyday mathematics, engineering mathematics, academic mathematics, as well as ethnomathematics see, for instance, Skovsmose , a , c ; Yasukawa et al. Thus Frankenstein emphasises the importance of respecting student knowledge. A foreground is defined through very many parameters having to do with: economic conditions, social-economic processes of inclusion and exclusion, cultural values and traditions, public discourses, racism.
This concern is, for instance, expressed through an intentionality-interpretation of meaning Skovsmose b. It becomes important to consider the space of possibilities that might be left open by this logic. These considerations have to do with the micro-macro classroom-society analyses as in particular addressed by Paola Valero see, for instance, Valero Society can be changed. This is the most general claim made in politics. It is the explicit claim of any activism. And it is as well a concern of critical mathematics education.
However, a warning has been formulated: one cannot talk about making socio-political changes without acknowledge the conditions for making changes see, for instance, Pais Thus the logic of schooling could obstruct many aspirations of critical mathematics education. Anyway, I find that it makes good sense to articulate a mathematics education for social justice, not least in a most unjust society. Notions such as social justice , mathemacy , dialogue , pedagogical imagination , and uncertainty are important for formulating concerns of critical mathematics education.
Social justice. Critical mathematics education includes a concern for addressing any form of suppression and exploitation. As already indicated, there is no guarantee that an educational approach might in fact be successful in bringing about any justice. Still, working for social justice is a principal concern of critical mathematics education. Addressing equity also represents concerns of critical mathematics education, and the discussion of social justice and equity bring us to address processes of inclusion and exclusion.
It is a concern of critical mathematics education to address any form of social exclusion see, for instance, Martin However, social inclusion might also represent a questionable process: it could mean an inclusion into the capitalist mode of production and consumption. So, critical mathematics education needs to address inclusion-exclusion as contested processes. However, many forms of inclusion-exclusion have until now not been discussed profoundly in mathematics education: the conditions of blind students, deaf students, students with different disabilities—in other words: students with particular rights.
However, such issues are now being addressed in the research environment created by the Lulu Healy and Miriam Goody Penteado in Brazil. Such initiatives bring new dimensions to critical mathematics education. Mathemacy is closely related to literacy, as formulated by Freire, being a competence in reading and writing the world. Chronaki provides a multifaceted interpretation of mathemacy, and in this way it is emphasised that this concept needs to be re-worked, re-interpreted, and re-developed in a never-ending process.
Different other notions have, however, been used as well for these complex competences, including mathematical literacy and mathematical agency. Jablonka provides a clarifying presentation of mathematical literacy, showing how this very notion plays a part in different discourses, including some that hardly represent critical mathematics education. The notion of mathematical agency helps to emphasise the importance of developing a capacity not only with respect to understanding and reflection but also with respect to acting.
Not least due to the inspiration from Freire, the notion of dialogue has played an important role in the formulation of critical mathematics education. Dialogic teaching and learning concerns forms of interaction in the classroom. It can be seen as an attempt to break at least some features of the logic of schooling and as a way of establishing conditions for developing mathemacy or mathematical literacy, or mathematical agency. Problem - based learning and project work can also be seen as way of framing a dialogic teaching and learning.
Critique in terms of imagination has been formulated in terms of sociological imagination. Through such imagination one reveals that something being the case could be different. I find that a pedagogical imagination makes part of a critical educational endeavour.
Such an imagination helps to show that alternative can be explored and that different possibilities might be within reach. I find that researching possibilities makes integral part of a critical mathematics education see, Skovsmose b , b ; Skovsmose and Penteado Critique cannot be any dogmatic exercise, in the sense that it can be based on any well-defined foundation.
One cannot take as given any particular theoretical basis for critical mathematics education; it is always in need of critique see, for instance, Ernest In particular one cannot assume any specific interpretation of social justice, mathemacy, dialogue, etc. They are all contested concepts. They are under construction. The open nature of critical mathematics education is further emphasised by the fact that forms of exploitations, suppressions, environmental problems, critical situations in general are continuously changing.
Critique cannot develop according to any pre-set programme. As a consequence, the basic epistemic condition for a critical activity is uncertainty see, Skovsmose a , b. Looking into the future much more is on its way. Let me just refer to some doctoral studies in progress that I am familiar with.
Ana Carolina Faustino addresses dialogical processes in primary mathematics education. Not least inspired by the work of Freire, dialogic education has developed with many references to adult education. However, Faustino is going to rework the very conception of dialogue with particular reference to younger children. Amanda Queiroz Moura investigates mathematics education for deaf students. This brings her to address particular aspects of inclusive education as well as of dialogical education, and in this way to provide new dimensions to the discussion for mathematics education for social justice.
Thus Muzinatti formulates new concerns of critical mathematics education. Guilherme Henrique Gomes da Silva addresses affirmative actions. He investigates the different components of such actions, emphasising that affirmative actions also must address the very educational format of university studies. This way da Silva brings a new specificity to the discussion of affirmative actions.
We always have to remember that much research in mathematics education is not presented in English, but in other languages. The same applies to critical mathematics education. Here one finds important contribution in Portuguese as, for instance, Biotto Filho , Marcone and Milani See also, Skovsmose , , ; Valero and Skovsmose Critical mathematics education is an on-going endeavour. Thus, when doing mathematics, we can see ourselves as telling a sort of story, talking as if numbers existed.
By this account, there are no metaphysical or epistemological problems special to mathematics. The only worries left are the general worries about non-mathematical physics, and about fiction in general. Field's approach has been very influential, but is widely rejected. This is in part because of the requirement of strong fragments of second-order logic to carry out his reduction, and because the statement of conservativity seems to require quantification over abstract models or deductions. Social constructivism see mathematics primarily as a social construct , as a product of culture, subject to correction and change.
Like the other sciences, mathematics is viewed as an empirical endeavor whose results are constantly evaluated and may be discarded. However, while on an empiricist view the evaluation is some sort of comparison with "reality", social constructivists emphasize that the direction of mathematical research is dictated by the fashions of the social group performing it or by the needs of the society financing it.
However, although such external forces may change the direction of some mathematical research, there are strong internal constraints—the mathematical traditions, methods, problems, meanings and values into which mathematicians are enculturated—that work to conserve the historically-defined discipline. This runs counter to the traditional beliefs of working mathematicians, that mathematics is somehow pure or objective. But social constructivists argue that mathematics is in fact grounded by much uncertainty: as mathematical practice evolves, the status of previous mathematics is cast into doubt, and is corrected to the degree it is required or desired by the current mathematical community.
This can be seen in the development of analysis from reexamination of the calculus of Leibniz and Newton.
Philosophy of Mathematics
They argue further that finished mathematics is often accorded too much status, and folk mathematics not enough, due to an overemphasis on axiomatic proof and peer review as practices. The social nature of mathematics is highlighted in its subcultures. Major discoveries can be made in one branch of mathematics and be relevant to another, yet the relationship goes undiscovered for lack of social contact between mathematicians. Social constructivists argue each speciality forms its own epistemic community and often has great difficulty communicating, or motivating the investigation of unifying conjectures that might relate different areas of mathematics.
Social constructivists see the process of "doing mathematics" as actually creating the meaning, while social realists see a deficiency either of human capacity to abstractify, or of human's cognitive bias , or of mathematicians' collective intelligence as preventing the comprehension of a real universe of mathematical objects. Social constructivists sometimes reject the search for foundations of mathematics as bound to fail, as pointless or even meaningless. Contributions to this school have been made by Imre Lakatos and Thomas Tymoczko , although it is not clear that either would endorse the title.
Reuben Hersh has also promoted the social view of mathematics, calling it a "humanistic" approach,  similar to but not quite the same as that associated with Alvin White;  one of Hersh's co-authors, Philip J. Davis , has expressed sympathy for the social view as well. A criticism of this approach is that it is trivial, based on the trivial observation that mathematics is a human activity [ citation needed ]. To observe that rigorous proof comes only after unrigorous conjecture, experimentation and speculation is true, but it is trivial and no-one would deny this. So it's a bit of a stretch to characterize a philosophy of mathematics in this way, on something trivially true.
The calculus of Leibniz and Newton was reexamined by mathematicians such as Weierstrass in order to rigorously prove the theorems thereof. There is nothing special or interesting about this, as it fits in with the more general trend of unrigorous ideas which are later made rigorous. There needs to be a clear distinction between the objects of study of mathematics and the study of the objects of study of mathematics.
The former doesn't seem to change a great deal; [ citation needed ] the latter is forever in flux. The latter is what the social theory is about, and the former is what Platonism et al. 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.
These objects, it asserts, are primarily semiotic objects existing in the sphere of human culture, sustained by social practices after Wittgenstein that utilize physically embodied signs and give rise to intrapersonal mental constructs. Social constructivists view the reification of the sphere of human culture into a Platonic realm, or some other heaven-like domain of existence beyond the physical world, a long-standing category error.
Rather than focus on narrow debates about the true nature of mathematical truth , or even on practices unique to mathematicians such as the proof , a growing movement from the s to the s began to question the idea of seeking foundations or finding any one right answer to why mathematics works. The starting point for this was Eugene Wigner 's famous paper The Unreasonable Effectiveness of Mathematics in the Natural Sciences , in which he argued that the happy coincidence of mathematics and physics being so well matched seemed to be unreasonable and hard to explain.
Realist and constructivist theories are normally taken to be contraries. In one sense it is irrefutable and logically true. In the second sense it is factually true and falsifiable. Another way of putting this is to say that a single number statement can express two propositions: one of which can be explained on constructivist lines; the other on realist lines. Innovations in the philosophy of language during the 20th century renewed interest in whether mathematics is, as is often said, the language of science. Although some mathematicians and philosophers would accept the statement " mathematics is a language ", linguists believe that the implications of such a statement must be considered.
For example, the tools of linguistics are not generally applied to the symbol systems of mathematics, that is, mathematics is studied in a markedly different way from other languages. If mathematics is a language, it is a different type of language from natural languages. Indeed, because of the need for clarity and specificity, the language of mathematics is far more constrained than natural languages studied by linguists.
However, the methods developed by Frege and Tarski for the study of mathematical language have been extended greatly by Tarski's student Richard Montague and other linguists working in formal semantics to show that the distinction between mathematical language and natural language may not be as great as it seems. Mohan Ganesalingam has analysed mathematical language using tools from formal linguistics. One important difference is that mathematical objects have clearly defined types , which can be explicitly defined in a text: "Effectively, we are allowed to introduce a word in one part of a sentence, and declare its part of speech in another; and this operation has no analogue in natural language.
This argument, associated with Willard Quine and Hilary Putnam , is considered by Stephen Yablo to be one of the most challenging arguments in favor of the acceptance of the existence of abstract mathematical entities, such as numbers and sets. The justification for the first premise is the most controversial. Both Putnam and Quine invoke naturalism to justify the exclusion of all non-scientific entities, and hence to defend the "only" part of "all and only".
The assertion that "all" entities postulated in scientific theories, including numbers, should be accepted as real is justified by confirmation holism. Since theories are not confirmed in a piecemeal fashion, but as a whole, there is no justification for excluding any of the entities referred to in well-confirmed theories. This puts the nominalist who wishes to exclude the existence of sets and non-Euclidean geometry , but to include the existence of quarks and other undetectable entities of physics, for example, in a difficult position.
The anti-realist " epistemic argument" against Platonism has been made by Paul Benacerraf and Hartry Field. Platonism posits that mathematical objects are abstract entities. By general agreement, abstract entities cannot interact causally with concrete, physical entities "the truth-values of our mathematical assertions depend on facts involving Platonic entities that reside in a realm outside of space-time" .
Whilst our knowledge of concrete, physical objects is based on our ability to perceive them, and therefore to causally interact with them, there is no parallel account of how mathematicians come to have knowledge of abstract objects. Field developed his views into fictionalism. Benacerraf also developed the philosophy of mathematical structuralism , according to which there are no mathematical objects. Nonetheless, some versions of structuralism are compatible with some versions of realism. The argument hinges on the idea that a satisfactory naturalistic account of thought processes in terms of brain processes can be given for mathematical reasoning along with everything else.
One line of defense is to maintain that this is false, so that mathematical reasoning uses some special intuition that involves contact with the Platonic realm. A modern form of this argument is given by Sir Roger Penrose. Another line of defense is to maintain that abstract objects are relevant to mathematical reasoning in a way that is non-causal, and not analogous to perception. This argument is developed by Jerrold Katz in his book Realistic Rationalism. A more radical defense is denial of physical reality, i.
In that case, a mathematician's knowledge of mathematics is one mathematical object making contact with another. Many practicing mathematicians have been drawn to their subject because of a sense of beauty they perceive in it.
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One sometimes hears the sentiment that mathematicians would like to leave philosophy to the philosophers and get back to mathematics—where, presumably, the beauty lies. In his work on the divine proportion , H. Huntley relates the feeling of reading and understanding someone else's proof of a theorem of mathematics to that of a viewer of a masterpiece of art—the reader of a proof has a similar sense of exhilaration at understanding as the original author of the proof, much as, he argues, the viewer of a masterpiece has a sense of exhilaration similar to the original painter or sculptor.
Indeed, one can study mathematical and scientific writings as literature. Philip J. Davis and Reuben Hersh have commented that the sense of mathematical beauty is universal amongst practicing mathematicians. The first is the traditional proof by contradiction , ascribed to Euclid ; the second is a more direct proof involving the fundamental theorem of arithmetic that, they argue, gets to the heart of the issue. Davis and Hersh argue that mathematicians find the second proof more aesthetically appealing because it gets closer to the nature of the problem.
There is not universal agreement that a result has one "most elegant" proof; Gregory Chaitin has argued against this idea. Philosophers have sometimes criticized mathematicians' sense of beauty or elegance as being, at best, vaguely stated. By the same token, however, philosophers of mathematics have sought to characterize what makes one proof more desirable than another when both are logically sound.
Another aspect of aesthetics concerning mathematics is mathematicians' views towards the possible uses of mathematics for purposes deemed unethical or inappropriate. The best-known exposition of this view occurs in G. Hardy 's book A Mathematician's Apology , in which Hardy argues that pure mathematics is superior in beauty to applied mathematics precisely because it cannot be used for war and similar ends.
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Main article: Formalism philosophy of mathematics. Main articles: Conventionalism and Preintuitionism. Main article: Mathematical intuitionism. Main article: Constructivism philosophy of mathematics. Main article: Finitism. Main article: Mathematical structuralism. Main article: Aristotle's theory of universals. See also: In re structuralism and Immanent realism. Main article: Psychologism. See also: Anti-psychologism. Main articles: Quasi-empiricism in mathematics and Postmodern mathematics. See also: Fictionalism. Main article: Social constructivism. Main article: Language of mathematics.
Axiomatic set theory Axiomatic system Category theory Definitions of mathematics Formal language Formal system Foundations of mathematics Golden ratio History of mathematics Intuitionistic logic Logic Mathematical beauty Mathematical constructivism Mathematical logic Mathematical proof Mathematicism Metamathematics Model theory Naive set theory Non-standard analysis Philosophy of language Philosophy of science Philosophy of physics Philosophy of probability Proof theory Rule of inference Science studies Scientific method Set theory The Unreasonable Effectiveness of Mathematics in the Natural Sciences Truth.
University of Exeter. Retrieved 28 March Introduction to Metamathematics. Reprinted, pp. Hart ed. Platonized Naturalism", The Journal of Philosophy , 92 10 : — Foundations of Physics. Bibcode : FoPh Putnam and G. Massey trans. Page Weber 's memorial article, as quoted and translated in Gonzalez Cabillon, Julio Retrieved Gonzalez gives as the sources for the memorial article, the following: Weber, H: "Leopold Kronecker", Jahresberichte der Deutschen Mathematiker Vereinigung , vol ii , pp.
Philosophers of mind interacted with psychologists and computer scientists to forge cognitive science, the new science of the mind. Philosophers of biology struggled with methodological issues related to evolution and the burgeoning field of genetics, and philosophers of physics worried about the coherence of the fundamental assumptions of quantum mechanics and general relativity.
Meanwhile, philosophers of mathematics were chiefly concerned with the question as to whether numbers and other abstract objects really exist. This fixation was not healthy. It has almost nothing to do with everyday mathematical practice, since mathematicians generally do not harbour doubts whether what they are doing is meaningful and useful — and, regardless, philosophy has had little reassurance to offer in that respect.
Insofar as it is possible to provide compelling justification for doing mathematics the way we do, it will not come from making general pronouncements but, rather, undertaking a careful study of the goals and methods of the subject and exploring the extent to which the methods are suited to the goals. When we begin to ask why mathematics looks the way it does and how it provides us with such powerful means of solving problems and explaining scientific phenomena, we find that the story is rich and complex.
The problem is that set-theoretic idealisation idealises too much. Mathematical thought is messy. When we dig beneath the neatly composed surface we find a great buzzing, blooming confusion of ideas, and we have a lot to learn about how mathematics channels these wellsprings of creativity into rigorous scientific discourse. But that requires doing hard work and getting our hands dirty. And so the call of the sirens is pleasant and enticing: mathematics is set theory! Just tell us a really good story about abstract objects, and the secrets of the Universe will be unlocked.
This siren song has held the philosophy of mathematics in thrall, leaving it to drift into the rocky shores.
WHAT IS THE PHILOSOPHY OF MATHEMATICS EDUCATION?
Given that the philosophy of mathematics has been closely aligned with logic for the past century or so, one would expect the fortunes of the two subjects to rise and fall in tandem. Over that period, logic has grown into a bona-fide branch of mathematics in its own right, and in Paul Cohen won a Fields Medal, the most prestigious prize in mathematics, for solving two longstanding open problems in set theory.
Ideally, information should flow back and forth, with philosophical understanding informing implementation and practical results, and challenges informing philosophical study. So it makes sense to consider the role that logic has played in computer science as well. A rival approach, with its origins in the s, incorporates neural networks, a computational model whose state is encoded by the activation strength of very large numbers of simple processors connected together like neurons in the brain.
The early decades of AI were dominated by the logic-based approach, but in the s researchers demonstrated that neural networks could be trained to recognise patterns and classify images without a manifest algorithm or encoding of features that would explain or justify the decision. This gave rise to the field of machine learning. Improvements to the methods and increased computational power have yielded great success and explosive growth in the past few years. The approach, known as deep learning, is now all the rage. Logic has also lost ground in other branches of automated reasoning.
Logic-based methods have yet to yield substantial success in automating mathematical practice, whereas statistical methods of drawing conclusions, especially those adapted to the analysis of extremely large data sets, are highly prized in industry and finance. Computational approaches to linguistics once involved mapping out the grammatical structure of language and then designing algorithms to parse down utterances to their logical form.
These days, however, language processing is generally a matter of statistical methods and machine learning, which underwrite our daily interactions with Siri and Alexa. The structure of language is inherently amorphous. Concepts have fuzzy boundaries. Whereas mathematics seeks precise and certain answers, obtaining them in real life is often intractable or outright impossible.
In such circumstances, what we really want are algorithms that return reasonable approximations to the right answers in an efficient and reliable manner. Real-world models also tend to rely on assumptions that are inherently uncertain and imprecise, and our software needs to handle such uncertainty and imprecision in robust ways.
Evidence for a scientific theory is rarely definitive but, rather, supports the hypotheses to varying degrees. If the appropriate scientific models in these domains require soft approaches rather than crisp mathematical descriptions, philosophy should take heed. We need to consider the possibility that, in the new millennium, the mathematical method is no longer fundamental to philosophy. But the rise of soft methods does not mean the end of logic. Our conversations with Siri and Alexa, for instance, are never very deep, and it is reasonable to think that more substantial interactions will require more precise representations under the hood.
In an article in The New Yorker in , the cognitive scientist Gary Marcus provided the following assessment:. For some purposes, soft methods are blatantly inappropriate. If you go online to change an airline reservation, the system needs to follow the relevant policies and charge your credit card accordingly, and any imprecision is unwarranted. Computer programs themselves are precise artifacts, and the question as to whether a program meets a design specification is fairly crisp.
Getting the answer right is especially important when that software is used to control an airplane, a nuclear reactor or a missile launch site. Even soft methods sometimes call for an element of hardness. The question, then, is not whether the acquisition of knowledge is inherently hard or soft but, rather, where each sort of knowledge is appropriate, and how the two approaches can be combined. Leslie Valiant, a winner of the celebrated Turing Award in computer science, has observed:.
Give us serenity to accept the things we cannot understand, courage to analyse the things we can, and wisdom to know the difference. W hat about the role of mathematical thought, beyond logic, in our philosophical understanding? The influence of mathematics on science, which has only increased over time, is telling.
Even soft approaches to acquiring knowledge are grounded in mathematics.
Aeon for Friends
Statistics is built on a foundation of mathematical probability, and neural networks are mathematical models whose properties are analysed and described in mathematical terms. To be sure, the methods make use of representations that are different from conventional representations of mathematical knowledge. But we use mathematics to make sense of the methods and understand what they do. Mathematics has been remarkably resilient when it comes to adapting to the needs of the sciences and meeting the conceptual challenges that they generate.
The world is uncertain, but mathematics gives us the theory of probability and statistics to cope. Newton solved the problem of calculating the motion of two orbiting bodies, but soon realised that the problem of predicting the motion of three orbiting bodies is computationally intractable.