If you are not familiar with the Philosophy of Mathematics Education Journal, see the link below.

http://people.exeter.ac.uk/PErnest/

If you are not familiar with the Philosophy of Mathematics Education Journal, see the link below.

http://people.exeter.ac.uk/PErnest/

Thank you Mr. Jones!

 Thanks!

This book is from 2007 but is still interesting since it aims to explore recent (at that time) development in this area including future suggestion for future research: Philosophical Dimensions in Mathematics Education: Balance between philosophy of and philosophy in mathematics education – and how to implement the former into the latter

Editors: Francois, Karen, Bendegem, Jean Paul Van (Eds.)

Thank you Ms. Sumpter!

Hello Luis,

This is a recently edited article that might be useful:

http://stanford.library.usyd.edu.au/entries/philosophy-mathematics/

Other potentially useful sources are:

https://faculty.unlv.edu/jwood/unlv/Articles/PlatoSelections.pdf

and the following books:

New Directions in the Philosophy of Mathematics: An Anthology

By Thomas Tymoczko

and

Philosophy of Mathematics: Selected Readings

By Paul Benacerraf, Hilary Putnam

Regards,

Daniela

Thank you, Ms. Vandepeer! I am going to read them!

I suggest one more book: How Humans learn to think mathematically by David Tall, Cambridge Univ Press

Guy

Thank you Guy!

Yes, I agree that this would be a good venue, this UK based journal is well established:

http://people.exeter.ac.uk/PErnest/

But I would also recommend this series

http://eric.ed.gov/?id=ED381351

The Philosophy of Mathematics Education. Studies in Mathematics Education Series: 1.

Ernest, Paul

Thank you, Miss Toda! Greetings of Peru!

 There is a Philosophy of Mathematics Education group (TSG 53) at ICME 13 Hamburg July 2016 - we have put out a call for papers 

The group rationale provides a partial answer to your question, Luis!

We are seeking 4 page submitted papers (8) for presentation at the conf in the study group.  Please contact me if you are interested in submitting.

TSG 53       Philosophy of mathematics education        

Co-chairs:

Paul Ernest (UK)                                          [email protected]

Ladislav Kvasz (Czech Republic)                       [email protected]

Team members:

Maria Bicudo (Brazil)

Regina Möller (Germany)

Ole Skovsmose (Denmark/Brazil)

IPC Liaison person: Susanne Prediger (Germany)

ABSTRACT

What is the philosophy of mathematics education? It can be an explicit position that is formulated, reformulated, criticized, refined, etc. But it can also refer to implicit assumptions and priorities, including paradigmatic assumptions that one need not be aware of, but which might be identified through, let us call it, a philosophical archaeology.

The philosophy of any activity comprises its aims or rationale. Thus we ask: what is the purpose of teaching and learning mathematics? An answer explains why we engage in these practices and what we hope will be achieved. But just considering such purposes quickly leads to seeing the divergence in aims and values of different groups.

A broader view looks at the applications of philosophy to mathematics education including topics such as epistemology, philosophy of mathematics, ethics and aesthetics. It applies philosophical methods to a critical examination of the assumptions, reasoning and conclusions of mathematics education, systematically enquiring into fundamental questions:

What is mathematics?

How does mathematics relate to society?

Why teach mathematics?

What is the nature of learning (mathematics)?

What is the nature of mathematics teaching?

What is the significance of information and communication technology in the teaching and learning of mathematics?

What is the status of mathematics education as knowledge field?

What deep and often unacknowledged assumptions underlie mathematics education research and practice?

Ethics is a central branch of philosophy that is often ignored or regarded as irrelevant for mathematics. What is or should be the role of ethics in mathematics education?

The philosophy of mathematics education matters because it gives people new ‘glasses' through which to see the world. It enables people to see beyond official stories about the society, mathematics, and education. It provides thinking tools for questioning the status quo, for seeing 'what is' is not what 'has to be'; enabling us to imagine alternatives possibilities. This is important throughout mathematics education research but also especially important in mathematics teacher education, when new mathematics teachers learn how to view the worlds of the teaching and learning of mathematics. 

The sessions will offer expert presentations (6) on key questions and issues of the field with plenty of space and time for questions, discussion and participation. It will also includes shorter presentations (8). Both of these will allow new issues to emerge to stimulate discussion and controversy, ultimately to encourage growth in research and teaching developments in mathematics education inspired by philosophical perspectives.

Thank you, Prof. Ernest! I'm going send a letter (by RG)  in next hours!

Dear colleagues:

Good morning. Dou you know marxists approaches on Philosophy of Mathematics Education?

Thank you! Greetings of Peru!

The philosophy of mathematics education has been dominated by Paul Ernest who is reluctant to respond to criticism, which puts doubt as to whether his 'philosophy' is in fact philosophy at all, since it is the very nature of philosophy to put forward an argument to have it criticised. Ernest's 'philosophy' seems little more than an undemocratic faith. There are quite a few papers criticising Ernest that have yet to receive a response. One such is:

Rowlands, S. Graham, T. and Berry, J. (2011), Problems with Fallibilism as a Philosophy of Mathematics Education, SCIENCE & EDUCATION 20(7):625-654 · JANUARY 

Hi Stuart, I haven't seen your 2011 paper. Thanks for the reference. I'd be delighted to read it and respond (if there is an appropriate place to do so). But actually I always welcome critique and discussion, indeed, it lies at the heart of my philosophy as well as my professional practice. With regard to your claim about faith - all of our arguments inevitably begin from unjustified assumptions (in the epistemological sense). Quite probably your assumptions differ from my assumptions, but we can still look at each others' reasoning and analysis, as well as critiquing each others assumptions. I know I benefit from this, and you might too!

Currently I've been exploring Plato's philosophy of mathematics education which is fascinating, especially since he argues that mathematics itself, as well as mathematics education, is ethical!  

I'm co-organizing the Topic Study Group on the philosophy of mathematics education (TSG53) at ICME13 in Hamburg summer 2016. You (and other readers of this) are welcome to submit a paper to the group. The quality of the submission - as opposed agreeing with my ideas is - the criterion for acceptance!  You are also invited to submit a paper for  Philosophy of Mathematics Education Journal issue number 30 (2016). - indeed I warmly invite you, Stuart, to submit a paper for this issue! If you wish I can write a response.

Topic Study Group (No. 53) on Philosophy of Mathematics Education at the 13th International Congress on Mathematical Education, July 24 - 31, 2016, Hamburg, Germany, for which papers are sought.

See http://www.icme13.org/topic_study_groups and http://www.icme13.org/files/tsg/TSG_53.pdf for details and please contact me directly if you would like to submit a paper or be involved in some way. [email protected]

Philosophy of Mathematics Education Journal, No. 29 July 2015 (25th Anniversary Issue) 

I am pleased to announce the publication of this issue of the journal via the website

http://people.exeter.ac.uk/PErnest/

The contents of the issue are listed below. The next issue will be published in 2016 and submissions are sought for it! Please start planning now and contact me if you wish to discuss any ideas with me (in the next 6+ months or so).

Philosophy of Mathematics Education Journal No. 29 (July 2015)

25th Anniversary Issue

Philosophy of Mathematics Education Journal ISSN 1465-2978 (Online)

Editor: Paul Ernest

CONTENTS

Paul Dowling Social Activity Method: A Fractal Language for Mathematics

David W. Stinson & Erika C. Bullock Critical Postmodern Methodology in Mathematics Education Research: Promoting another way of Thinking and Looking

Florian Schacht & Stephan Hussmann Between the Social and the Individual: Reconfiguring a Familiar Relation

Galit Caduri & Einat Heyd-Metzuyanim Is Collaboration Across Incommensurable Theories in Mathematics Education Possible?

Heather Mendick Using Popular Culture to Teach Maths

Elizabeth Jackson Student Primary Teachers' Perceptions of Mathematics

Paul Ernest Problem Solving: Its Assimilation to the Teacher's Perspective

Barbara Pieronkiewicz Affective Transgression as the Core Objective of Mathematics Education

Laura Golnabi The Conditions of Flow and Mathematical Problem Solving

Inês Hipolito What is a Mathematical Proof in the Age of Modern Theorem Provers

Andreas Moutsios-Rentzos & Panagiotis Spyrou Fostering internal need for proof: A reading of the genesis of proof in ancient Greece

Michael DeVilliers I Have a Dream and Crossed Quadrilaterals

Sean Chorney The Practice of Learning Mathematics as Comprising Three Parallel Processes: Mathematics, Tool and Student

Regina Dorothea Moeller Teaching the Concept of Velocity in Mathematics Classes

Anna Bellamy A Critical Analysis of Teaching and Learning Negative Numbers

Jeffrey David Pair Is God a Mathematician?

Douglas Henrich Mathematical Ethics: Values, Valences and Virtue

Mario A. Natiello and Hernán G. Solari On the Removal of Infinities from Divergent Series

----------------------------------------

I am very gratefull with your answers, Prof. Rowlands and Prof. Ernest. If you wish it, you can write in this thread. Only, I would like to know about my last question: are there (actually) Marxists (or Dialectical Materialists) approaches on Philosophy of Mathematics Education?

Greetings of Peru!

Dear George:

Thank you for your answer. Precisely, I am researching about this topic. I don't believe that 70 years of USSR and another socialist countries (and still I don't have mentioned the Marxist thought in Western Hemisphere) don't produce materials on Philosophy of Mathematics. 

Best Regards!

Thank you Paul for responding to my answer. I co-authored the above reference (Problems with Fallibilism as a Philosophy of Mathematics Education) because of the lack of response to the article:

 Rowlands, S., Graham, E. and Berry, J.: 2001, 'An Objectivist Critique of Relativism in Mathematics Education', Science & Education, 10(3), 215-241

If you remember Paul, I emailed you this reference when you emailed me concerning the article by Rowlands and Carson in your PoME journal number 14 (you made the point that my criticism of constructivism didn't include social constructivism - so I made the reference).

Rowlands, Graham and Berry (2011) is essentially a repeat of Rowlands, Graham and Berry (2001) and yet I am not aware of any response to this article written 14 years ago. Since you welcome debate I shall look forward to your published response to the two articles.  I don't need to write any more on this, or attend your conference session, until you have responded to these two papers. I'm afraid the ball is in your court.

I am very surprised at your forgetfulness, since you seem to be well up on current trends concerning your philosophy of mathematics education and you seem to respond very quickly to these trends. The irony is that the criticism of your PoME is very rare indeed. 

I shall look forward to your response 

Dear Stuart - I looked at both references yesterday and today. Yes, I recall the 2001 paper although I had not seen the 2011 paper. These are rather long papers which choose certain aspects of my work to criticise, as well as critiquing broader trends (well -groupings that you construct) in developments in the PoME. It would be a very long job indeed to go through the papers and offer a critique, and I'm not sure of anywhere other than the PoME journal that would print such a response. Some of the claims you make are unjustified, some are anecdotal, some depend on cherry-picking quotes from a range of authors,  but more importantly there are a sequence of philosophical points and criticisms in your (joint) paper(s) that are genuine issues worthy of discussion.

For example, you argue that it is not enough simply for social acceptance to establish the objectivity of mathematical knowledge claims. This is an issue I struggle with myself. Clearly getting votes is not enough to warrant theorems. You might say that they also need to be true. But for me that is circular, because I seek the warrant for truth claims, enabling me to claim they are true.

Is it enough for long entrenched rules of reasoning, logic and deduction (both explicit and tacit) – themselves the result of historical and social acceptance within the community of mathematicians--  to provide a warrant for mathematical knowledge claims (i.e., the acceptance of proofs by the community of mathematicians)?

Perhaps not – perhaps what is needed is a belief in the ideology of objectivism applied to mathematical proof and its results. Thus maybe it is important that there appear to be absolute and objective criteria for a warranted mathematical theorem for mathematicians and others to believe in the certainty of mathematical knowledge. But note I am considering an illusion of objectivity, rather than its reality.

Even if the criteria are socially objective as opposed to universally or absolutely objective (a distinction I make at length elsewhere) – perhaps a belief in the latter is needed to treat the warrants with utmost respect.

These are simply my current unworked through speculations. Of course accounting for mathematical warrants in a socially constructivist PM is a tricky business. (But note Julian Coles' recent contributions in this area.) A traditional objectivist account is a lot simpler to maintain. But I believe that claiming that the human mind (and culture) can come up with objective distinctions and truths which would compel the assent of any sentient being in any universe is arrogant and an expression of hubris for us humans -  we are just an animal species that has evolved recently in one little corner of one galaxy!. What right have we to claim to own absolute truth?

Now Stuart, you are challenging me to address your critiques. If you offer me a version of your 2001 or 2011 paper(s) or a selection thereof for publication (even a simple reprint if you can get permission)  I am willing to write and publish a response  alongside your paper in PoME journal. I would be doing this for a wider audience, not just for you personally, as I am doing here. My reply would require considerable work. If you can't be bothered to submit a paper, so that the journal readers can follow the dialogue, don't expect me to either, unless these issues come up independently for me. If you do participate I could let you see my reply for your possible riposte. Over to you!

Dear Ernest,

I am so glad that you have, in fact, read my first article and that, amongst all the "cherry picking", "anecdotes" etc. of that article you managed to find issues worthy of discussion. If I can remind you of the central issue of that article:  There is a big contradiction between the strong claim in your first book that it is only social acceptance that confers objectivity (with logical necessity subsumed under the category of social acceptance) with the considerable weaker claim in your second book that logical necessity is over and above social acceptance (while still maintaining the second book is a continuation of the first).

Yes Paul, I can be "bothered" to submit  a paper to your journal, and I can be "bothered" to give conference  presentation (I am hoping to go to Hamburg in July), I was simply making the point that the ball is really in your court. Since now you seem eager for a debate, as I hoped you would be, I will submit the second article (I'll have to clear it with Springer first). for your perusal and with the view of publication in your journal. If Springer doesn't allow this then I will post the second article to you and write a condensed version for your journal.

As a taster, the second article, although a continuation of the first, centrally criticises your strong fallibilism (e.g.it is not the case that non-Euclidean geometry has falsified the angle property of the triangle, it has merely set the domain for the angle property to be true, i.e. plane triangles; and that Lakatos has not argued the strong position as you have asserted) and criticises your making synonymous epistemological issues with equity issues.

I hope that this debate will be published prior to discussion at Hamburg. 

Above you stated: "If you can't be bothered to submit a paper, so that the journal readers can follow the dialogue, don't expect me to either". Well, Paul, since I am "bothered", let the debates begin!

Sincerely,

Stuart

Excellent!  I look forward to the submission. One difference I noted between the 2001 and 2011 papers is that the second paper addresses (critiques) my pointing out the parallel (analogy) between proof and calculation. This is something that has not been stressed in the literature, and as I point out, the significance of calculation has been talked down and the significance of proof  has been talked  up. My claim is not that they are epistemologically equivalent, but to point out the ideological significance of trying to exaggerate the difference. Note that at the heart of Goedel's 1931 incompleteness theorems is the arithmetization of proof - expressing provability with arithmetical functions, ie through calculation! This is one of the most important results of the 20th century, but have not seen my point about the underlying analogy debated elsewhere so I welcome and look forward to trying to answer your criticisms.

With regard to changes in my philosophical position, I freely admit to this. It would be sad if I didn't learn and change and grow as a result of further thought and dialogue with others. For example, I now accept the certainty of mathematical knowledge but with the caveat that I distinguish social objectivity - including certainty in a humanly constrained way - from universal objectivity and absolutely certainty, which I continue to reject.  I have a paper on this appearing in Educational Studies in Mathematics (The problem of certainty). In my 1998 book (Social Constructivism as a Philosophy of Mathematics) I explicitly indicate at the beginning a number of ways in which my philosophy changed. 

Incidentally, your remarks would read better if you maintained an objective and philosophical tone. I'm sure that the substantial issues between us are what readers find most interesting. 

Dear Paul,

Since you have read my second paper then perhaps I don't need to send you a printed copy. I will still try, however, to obtain permission from Springer to publish the second paper in your journal.

I would like to state at the outset that Godel's proof is not a calculation but a systematic codification of formal logic using the statements of arithmetic. 

I think the best way to have an objective philosophical tone in any response is to be clear as to what is being debated here. For example, consider your statement above:

"I now accept the certainty of mathematical knowledge but with the caveat that I distinguish social objectivity - including certainty in a humanly constrained way - from universal objectivity and absolutely certainty, which I continue to reject."

What is social objectivity compared with universal objectivity? If someone creates a proof of a theorem that is valid, then surely the validity of that proof is universal - anyone who understands the language game at the required level can see that it is valid, prior to any consensus (or 'social objectivity'), despite the rejection of absolute certainty.  If a published proof of a theorem is shown to be invalid (such as the two proofs of the 4 - colour theorem in the 1890's) then the errors existed prior to their discovery and is therefore independent of 'social objectivity'. 

Of course, it would be much easier to maintain an objective philosophical tone if one does not refer to the other's intention (such as whether he or she is "bothered" to do something), or how badly someone has written their article . So let us stick to the formal and the objective with no ad hominem remarks. I am sure that with this spirit of discourse the debate will be long but very fruitful.  

Dear Paul, I cannot express how happy I am to have a public debate on the PoME. It is so long overdue. I'll shall look forward to the debate in Hamburg but prior to that I will try and get something published in your journal.

All the very best to you,

Stuart 

.

Excellent - do you mean that you will submit a paper to TSG53 (PoME) group in Hamburg to be debated or did I misunderstand? Or just attend?  I look forward to your submission to PoME Journal no 30 (2016). Best wishes - Paul

PS I attach a draft of my in press paper for ESM which explains what I mean by the difference between social objectivity and universal objectivity. The published version will have further modifications and corrections but is substantially the same.

Re Goedel's proof what he shows is that there is an arithmetical function (primitive recursive, ie using induction), lets call it g into which  you can put in two values T for theorem and P for proof (T and P belong to N, but encode the notions from Russell-Whitehead first order logic). Then the solution to the calculation g(T,P) shows that P proves T. I forget if it is actually g(T,P)=1. But it is a calculation, and is decidable. Whereas: There is an x s.t. g(T,x)=1 is not decidable.

Dear Paul,

Hopefully I shall present a paper at Hamburg rather than merely attend (it all depends on the available funding) but I shall make every effort to attend. I shall definitely submit to your journal, however.

Thank you for your article which I read with great interest; and indeed there is much that I agree with. I did, however, find it difficult to marry the article with your first book, which tends to downplay the certainty of mathematics, until I came to footnote 6 which explained the difference in your perspective.

May I humbly suggest that footnote 6 is incorporated in the main text since it does make a very important point concerning your change in perspective (and as you intimated above, it would be very odd if one didn’t change their outlook over 24 years). It is a fine article and I shall look forward to reading the final, polished and published version.

I’m in the process of appealing to Springer for publication in your journal, so I shall get back to you in the near future.

Best wishes,

Stuart