I would add structure and form as values of mathematics

Predictive power and accuracy of describing Nature. The recent discovery of the Higgs boson particle, which was predicted by mathematical formulas almost 50 years ago, shows the power of math to describe and predict the world around us.

I like both these replies - the appreciation of structure and form are part of the aesthetic values of mathematics

Another valued aspect is maths modelling power - as in particle physics's use of group theory etc

My tentative analysis of beauty and aesthetic values in mathematics is as follows:

A first analysis of some of the dimensions of mathematical beauty reported by mathematicians and in the literature, arrived at by the author in top down fashion, is as follows, listed in no particular order:

1. Economy, simplicity, brevity, succinctness, elegance.

The compression of a formula or a theorem of wide generality or an argument (proof) into a few short signs in mathematics is valued and admired.

2. Generality, abstraction, power.

The breadth and scope of a generality or a proof also evokes appreciation.

These two first criteria overlap somewhat.

3. Surprise, ingenuity, cleverness

Unexpectedness, like wit, is appreciated and valued when it reveals a new knowledge connection, as is the construction of hitherto unknown links.

4. Pattern, structure, symmetry, regularity, visual design

The discernment of pattern in its various and abstracted forms is the closest the values of mathematics come to those of art and general aesthetics in the visual field, although in mathematics these properties can be abstract.

5. Logicality, rigour, tight reasoning and deduction, pure thought

The development of logical reasoning to its ultimate forms of rigour and purity of thought is a valued part of mathematics and a well constructed mathematical proof evokes admiration like a gold necklace with well forged links.

6. Applicability, modelling power, empirical generality

Like metaphors in poetry the capture of empirical situations in mathematical models and more generally in applied theories and concepts is something appreciated both within and outside mathematics as a demonstration of the power and ‘unreasonable effectiveness’ of mathematics in the physical world (Wigner 1960), as opposed to the world of pure mathematics.

Paul Ernest (2013) ‘Values and Mathematics: Overt and Covert’, presented at Mathematical Cultures, Conference 2, September 17-19 2013, De Morgan House, London, Organiser: Brendan Larvor, University of Hertfordshire.

note that none of these values can be claimed as anything but personal and particular, but are being spoken (well, written...) as if universal. I think that may be an insuperable problem in any discussion of values.

Hi Donald - thanks for this - yes I struggled with this - but in my view values are cultural and not just personal - the are created by humanity but are not the idiosyncratic 'property' of individuals.

Here is what I say in the paper

Objective versus Subjective Values

In opening I referred to the widespread view that values only enter into mathematics through mathematicians’ subjectivity, and contrasted this with Plato’s view that there are a range of values that can be termed objective. This perspective has certain consequences, for if the values attributed to mathematics are purely subjective then they can be dismissed as personal responses to mathematics rather than an objective feature of the subject. However, if it can be shown that mathematics does have objective values then they must be taken more seriously as attributes of mathematics itself and not merely a feature of individuals’ relationship with mathematics. But there is an ambiguity in the word objective and I wish to contrast two different meanings. On the one hand, objectivity can refer to having an existence in the physical world as a brute fact verifiable by the senses or in the domain of knowledge by dint of logical necessity. On the other hand, objectivity can refer to the opposite of subjectivity. In this sense objectivity means having an existence that goes beyond any individual knower’s beliefs. These are not the same meanings, because mathematical objects can exist in the social and cultural realm beyond any individual beliefs without having independent physical existence or existence due to logical necessity. In this second sense objectivity is in effect redefined as social, as I argue in Ernest (1998a) drawing on the social theory of objectivity proposed by Bloor (1984), Harding (1986), Fuller (1988) and others. This is how social constructivism views mathematical objects. Such a perspective has a strong bearing on the discussion of values in mathematics because it posits that at least some of mathematics is contingent on human history and culture and thus it allows that mathematics itself can be imbued with the values of the culture of its human makers. Thus my overall claim is that mathematics can be understood as being laden with objective values from human culture, but that these are values that are not reducible to either subjective beliefs about mathematics or logically necessary properties of mathematics.

I believe that there is a dialectical relation between personal and cultural values. Certainly I want to claim that we internalize cultural values and they become personal values, appropriated and personalised int he process. I cannot easily describe how personal values become cultural values, but the process is part of the dialectical cyclic interdependency of the private/public and personal/cultural found in, say, Vygotsky. What we call the values of mathematics would be primarily cultural, but there are disagreements about what, say, beauty in mathematics is, which reveals a some personal elements - see the work of M Inglis. Does this answer your question?

Dear UM

I sort of agree with you about

"On this view individuals create variation and societies constrain it."

I am cagey because I think the individual/social dichotomy dissolves a bit - both are mutually constitutive. Clearly individuals sustain and recreate society and its variations, but our social contexts are deeply inscribed in us making us the individuals that we are in a very deep way (eg, our modes of thought are conttuted by the acquisition of language understood broadly)

My understanding (and elaboration) of Vygotsky's account of knowledge, language and indeed values appropriation and use (I add in the values bit for this discussion) can be represented by the following cycle - attached

 For my ref to Inglis see, eg, 

BEAUTY IS NOT SIMPLICITY: AN ANALYSIS OFMATHEMATICIANS’ PROOF APPRAISALS

MATTHEW INGLIS AND ANDREW ABERDEIN

https://www.academia.edu/6940192/Beauty_is_not_simplicity_An_analysis_of_mathematicians_proof_appraisals

You have to join Academia.Edu

I would suggest the following:

Context and related Values

Historical and Cultural

Doing your best- seek  to accomplish something worthy, try hard, pursue excellence

Respect – treat others with regard, listen other persons point of view

Socio –Political

Fair Go- protect the common good, treat all fairly for a just society

Honesty – seek the truth, be sincere

Integrity- act with  morality , ensure consistency

Individual

Care and Compassion- care for self and others

Responsibility- be accountable, take care  and contribute to society and civic life  

Hi Khalid

These are interesting and indeed commendable personal values. They do raise a couple of questions,such as who endoreses or proposes them (as well as you) and on what basis?

I think it is useful to distinguish between Proposed Values, Espoused Values and Enacted Values

What you have offered is a set of Proposed Values

In addition we might consider what are the Espoused Values of mathematicians (what they say), plus their Enacted Values (how they act in an ethical way) - to answer these questions we need empirical data

In addition there are the values embedded within mathematics itself. This is a bit harder to conceptually clarify as we as to ascertain. My contention is that Mathematics itself contains epistemological, ontological, aesthetic and  ethical values (re the latter see my dialogue published in Mathematical Intelligencer). However this is a controversial position!