The Mathematics of Mathematics: Thinking with the Late, Spinozist Vygotsky by Wolff-Michael Roth (2017)

The Preface and Chapter One is available on line

1 Vygotsky's Marxist-Spinozist Re/Orientation
2 The Thinking Body
3 The Mathematics of Mathematics
4 Sociogenesis
5 Intra-Intersubjective Field
6 Perezivanie - Experience
7 Affect and Emotion
8 Obucenie - Teaching / Learning
9 Mathematics in the Drama of Life
10 Overcoming Dualism

Vygotsky (in his previously unpublished notes) acknowledges the remnants of Cartesian dualism in his work, including an overemphasis of the intellectual over affect and the practical ...

To overcome these remants in his own work, Vygotsky turned to the philosopher Baruch Spinoza, who had postulated that there was on one substance that has body (Extension) and Thought as attributes. Accordingly, there are not two substances, body and thought (mind), biology and culture, or nature and nurture, but only one substance that manifests itself in different, mutually exclusive ways. This one substance is the thinking body, however, it is not the material human body: 'Thought can ... only be understood through investigation of its mode of action in the system thinking body - nature as a whole' ...

The most fundamental idea in The German Ideology is the primacy of social relations to anything that distinguishes humans from other species. Consciousness, the ideal, and the general all are societal in nature - not just as per their origin but also in their very existence. The ideal, such as a mathematical 'abstraction' or a mathematical 'idea', exists in the form of human relations. This insight led me to the title of this book, The Mathematics of Mathematics. That is, mathematics is not 'socially constructed' because humans have produced with others (ie. 'socially') and 'negotiated' some idea. In this way of approaching mathematics, 'the social' is incidental. In any event, any mathematical discovery is ascribed to individuals and, thus, is understood not socially: something like Fermat's Last Theorem is considered to have been an individual construction before it became social. This, however, is not consistent with how Marx or the late Vygotsky thought, where any higher psychological function was a relation with another person. Therefore, what is mathematical in mathematics is not merely (contingently) social but exists in the form of (universal) societal relations. Children learn mathematics and mathematical forms because they exist in public, that is, because these exist as relations in which the children are an integral part. Because, according to the Ideology, consciousness is conscious being, mathematics becomes individual when the child becomes conscious of the relation with others where the mathematical form exists as joint praxis....

these ideas are challenged by this book (existing awareness --> new awareness):
a) mediational nature of the sign (language) --> semiotic (sense-giving) speech field
b) meaning --> sense
c) the zone of proximal development --> the primacy of the social
d) thought --> unity / identity of intellect, affect and praxis
e) thinking --> thinking and speaking as two lines of development in communication
f) the distinction between intra and intersubjectivity (inside-outside) --> intra-inter subjectivity...

Episodes mostly from elementary school mathematics classrooms but some also involving scientists are used for developing and exemplifying the theory and method ...

Comment (BK): One challenge is how should we conceive / think about the individual, eg. Fermat, Marx, Spinoza etc. Didn't they have superior insights into society than other individuals? Shouldn't that aspect of their individuality be acknowledged? Their insights derived from the social but they were better at articulating what was happening than others. In this view of history individuals are important because individuals make breakthroughs and then those breakthroughs can be spread through painstaking education to the masses or society as a whole. We honour those individuals because we recognise their exceptional talents.

Reference to Preface:
Maxine Sheets-Johnstone, The Corporeal Turn: An Interdisciplinary Reader
Evald V Ilenkov, Dialectical Logic: Essays on it History and Theory
Marx & Engels, The German Ideology
Benedict de Spinoza, The Ethics

1 Vygotsky's Marxist-Spinozist Re/Orientation


Foundational Determinations

The difference between Rene Descarte (1596-1650) and Baruch Spinoza (1632-1677)

Descarte's ontology - there are two basic, incommensurable kinds of substances, body (extension) and soul (mind)

Spinoza - there is one substance, which constitutes Nature (Existence, Being) as a whole. This substance can't be grasped or seen as such. In thinking the substance is reduced to thought. In perception it is reduced to extension (body). Extension (body) and thought (ideas) are attributes of substance
No relation is possible across kinds so body can't influence thought and thought can't influence body.

Comment BK: How can we understand or accept substance if it is not material and can't be perceived, is not sensual? Is this mysticism from Spinoza? Is it a leap of faith on the part of Roth? ie. start from the POV that Descarte is wrong, that we can't accept dualism. This then leads to a leap of faith that there must be something which transcends the mind / body dualism, the mysterious "substance", which is Nature, Existence, Being as a whole. Personally, I want to believe too, since I don't like dualism, but still what I see as a leap of faith troubles me. I tentatively conclude that: I can accept that Nature, Existence, Being as a whole is an abstraction that can't be perceived.

There are other abstract phenomena such as 'worldview' or 'zeitgeist' or 'impressionism" which cannot be perceived. But they can be documented. A painting by Cezanne or a composition by Debussy are documentary manifestations of impressionism. These are different material manifestations of impressionism.

The manifestations of substance, thought and body are incommensurable and can't influence each other. An analogy is the duck-rabbit drawing. You can either see the duck or see the rabbit but you can't see both at the same time. The picture manifests itself in contradictory ways, duck and rabbit.

Duck-rabbit. Perception is active and ambiguous.

If body (biology) can't influence thought (consciousness) and consciousness can't influence biology then it follows that the modern neuropsychological studies where a MRI image of the brain is made while the human subject does something will not turn out to be successful in explaining how the brain works.

Comment BK: This seems to be a good empirical test of the Spinoza-Roth approach. I'm sure that those doing neuropsychological studies will have made claims that their approach is providing insights into thinking. Aside: I saw a report on TV about a paraplegic, no leg movement. They stimulated his toes and detected changes in the brain and concluded that the nerves connecting the toes to the brain were not broken. This raised hopes that somehow he should be able to move his toes.

Looking in the brain to explain thought or mind is looking in the wrong place because thought emerges, is an attribute of, an unobservable substance, the thinking body. The two forms of movement or change (brain changes - thought changes) are different manifestations or attributes of one and the same phenomena. They are not causally related.

Roth quotes Ilyenkov, Hegel,Vygotsky, Suchman and himself in support of this position.

Paraphrasing, for abbreviation and clarity:
Ilyenkov: it is impossible to understand thought through a study of the body or brain
Hegel critiqued the phrenologists who attempted to correlate bumps on the head with consciousness
Vygotsky: it is ridiculous to look for higher psychological functions in the cortex ... they must be explained ... in external terms
Suchman and Roth are cited for research that exhibits the gap between plans, forms of thoughts and situated material action

Comment BK: I found a free pdf of Suchman's book and a paper on line by Roth which seems to cover the same ground as the reference he cites. See References for this chapter below.

NB. The notion of teacher's 'scaffolding' students' construction of 'meaning' is challenged

Comment BK: I found another paper by Roth explaining this in more detail: Re/thinking the Zone of Proximal Development (Symmetrically) He argues that the teacher and student scaffold each other, eg. the student scaffolds the teacher when he reveals what he doesn't know either through a response to a teacher comment / question or through silence at a strategic moment

Comment BK: teacher directed methods like Direct Instruction do work but they do require the teacher to conscientiously listen to students and do the marking and then revisit those areas where students are not learning. In this way the students scaffold what the teacher does.

Transactional perspective (John Dewey and Gregory Bateson)

  • Self actional: the Self constructs itself
  • Interactional: two or more autonomous Selves relate to each other, each external to the other
  • Transactional: the Selves cannot be specified independently, they are interdependent parts of a whole, each reflecting the whole and, therefore, also the other

Comment BK: There is a good diagram illustrating these 3 models on p. 5 but I can't find a copy of it on the web

Thought is an attribute of Nature. Its infinite mind is the infinite intellect, whereas its finite mode is the individual mind.... the individual mind considers the whole partially, one sidedly and thus inadequately

How do we explain the relationship between thinking and the world?

Take the case of the hands of an expert typist. The hands and finger movements are shaped by the layout of the keyboard.

The real system is the thinking body in its context within which it does and thinks

The entire research field known as situated cognition understands this.

Comment BK: refer Andy Clark's book (enactivism) and Situated Learning entries

When we see the eyes caress, constantly move, over the body we see

The things in the environment determine our thoughts in a deep structural sense
Our hands:
  • type on the QWERTY keyboard
  • handle a knife and fork while eating
  • hold a pen while writing
  • tie shoelaces

Marx improves on Spinoza in pointing out that humans make things that in turn influence what humans become

Comment BK: cf McLuhan, the medium is the message

We comprehend the world because we are embedded in the world

The Self (individual) is the product of the social

The thinking body changes nature and hence itself, in a dynamic spiral

The thinking body has a practical sense, responding quickly in the heat of the situation, without having to stop and reflect. This is because it has been shaped by the world.

From Vygotsky's Readings of Spinoza

Vygotsky's monism replaces both dualist and parallelist takes on the mind-body relation. Other approaches end up creating a theory of unity and fail to establish a theory of identity of the psychological and the physical.

Speech having both a material (phonetic) and ideal (semantic) dimension is the key to the psychophysical (mind-body) problem. We can make an analogy between speech with its sensible-supersensible components and the commodity as analysed by Marx with its sensible exchange value and supersensible use value.

Vygotsky: "the curse of matter on pure consciousness is moving layers of air, ie. intercourse with the aid of language, rather than a connection to the brain!"

Speech is not the externalised result of a finished thought but thought becomes itself in speaking. Speech is material, bodily which is different from the semantic, ideal aspects (meanings) that are attributed to words. As speech unfolds, thinking develops so that the mind grasps its thinking only when speaking is finished.

Vygotsky and the Feuerbach-Marx Connection

Spinoza: Thought is an attribute of substance (Nature as a whole) rather than the individual mind

Marx: Thought, consciousness is a societal product

In an evolutionary sense deeds precede words

Being, Life determines consciousness

"The consciousness of sensation and thinking are characterised by different modes of reflecting reality. They are different types of consciousness. Therefore, thinking and speech are the key to understanding the nature of human consciousness. But if 'language is as ancient as consciousness itself', if 'language is consciousness that exists in practice for other people and therefore for myself', when 'the curse of matter, the curse of moving layers of air hangs over consciousness from the beginning', then it is not only the development of thought but the development of consciousness as a whole that is connected to the development of the word. Studies consistently demonstrate that the word plays a central role not in the isolated functions but the whole of consciousness. In consciousness, the word is what - in Feuerbach's words - is absolutely impossible for one person but possible for two. The word is the most direct manifestation of the historical nature of human consciousness"

What stands out from the Vygotsky quote:
That words are material, moving layers of air
Words are social, without others, there can be no words
Words create consciousness, as a whole

... society and societal relations are a condition for the emergence of consciousness in the form of language, which is an integral part of the material part of society, the real process of labour that is orientated to the satisfaction of basic and extended needs .... society is that unit that allows us to understand the individual; and as stated in the Eighth Thesis on Feuerbach, 'all societal life is essentially practical'.

Labour produces language

the nature and function of thinking, the intellectual needs, interests, inclinations and feelings of man are grounded in the material life of society

Some writings (classical social sciences) separate the intellectual from the material aspects of language, intellect is abstracted from the whole

thinking was divorced from life, from the living motives, interests and inclinations of the thinking person

the beginnings and ends of the chain of events that determine life are not located within the thinking body but outside it

Philosophy and psychology are about the whole person, not reason

Thinking is not a subject for itself but is a predicate of a real person

Idealism (abstraction) removes thinking from the real person, the real context

Truth is not theoretical but practical (Second Thesis on Feuerbach)

repeating from above and adding to it, Vygotsky: it is ridiculous to look for higher psychological functions in the cortex ... they must be explained ... in external terms, on the basis of the fact that man controls the activity of his brain from without through stimuli

explain the unicorn: Language is common to us, the unicorn exists as a word and image

Sixth Thesis on Feuerbach: human nature is not an abstraction inherent in the single individual. In its reality it is the ensemble of social relations

In contrast Feuerbach, arguing against religion, found the essence of man located in the individual, not seeing that the individual belongs to a particular form of society (Sixth and Seventh Thesis)

Society produces an ensemble of human relations; the ensemble of human relations produces society

NB. Roth critiques the constructivists and enactivists for making a similar error to Feuerbach, that humans are the results of their own constructions, end products of their actions, bearing the same characteristics of their actions

Vygotsky: any higher psychological function was first a real, social relation between two concrete human beings ... speech is both a social relation and a psychological means ... all forms of verbal communication between adult and child later become psychological functions

Georges Politzer conceived of psychology in terms of drama, a richer conception than the world of nature (science) or the world of the mind (psychology)
(developed wrt to mathematics in Ch 9)

Drama is neither internal or external, it always has a front and back stage and neither can be understood without the other

On Method
Covers primacy of practice, unit analysis, dialectics and methods for a concrete, human psychology of mathematics

Primacy of Praxis

Praxis is the starting point, not the object or intuition
Praxis: the actual, sensual labour through which things get done
Praxis: the full vitality of life: needs, interests, incentives and tendencies

Marx (1st thesis on Feuerbach): "The chief defect of all hitherto existing materialism -- that of Feuerbach included -- is that the thing [Gegenstand ], reality, sensuousness, is conceived only in the form of the object [Objekt ] or of intuition [Anschauung ],* but not as human sensuous activity, practice, not subjectively"

Concrete human psychology is concerned with how real living people, in the here and now of their concrete lives, make, and are made by, the everyday world they inhabit; concrete human psychology attempts to understand how people succeed in doing what they do without having a conception or representation thereof in their minds. Thus if we see a red light, we stop at the intersection. We do not see the light, interpret it, and then direct the relevant organ to push of pull the brakes.

Comment BK: my emphasis about the red light, since I find this interpretation remarkable

Marx: "it is not consciousness that determines life, but life that determines consciousness"

novel, unexpected things happen
what people actually do can be dramatic
when we do maths publicly, the maths is out there in the open and we do not have to get it into the head
the unfolding drama of social life explains consciousness

Comment BK: There is such a thing as internal knowledge construction. Why can't we have both: public maths and changes inside the head?

Idealism searches for categories and concepts and then fits events to those categories / concepts. Real life drama is degraded in texts.

A focus on meaning leads to the primacy of mind (Politzer). Marxist materialism focuses on real societal relations.

Marx: once material and mental labour are separated; "from this moment onwards consciousness can really flatter itself that it is something other than consciousness of existing practice, that it really represents something without representing something real"

Paul Ricoeur footnote, p. 22:
mimesis1: we make and observe an order while participating in social life
mimesis2: social life is transposed into the world of text
mimesis3: the world of text is transposed back into social life as part of the application of research results

In Structural theory entities remain stable. In real life things keep changing and moving.

Since human essence is societal relations then you can't reduce behaviour to the individual

Unit Analysis

Comment BK: Given that in Vygotsky the Unit of Analysis is not the individual then that goes a very long way to diminishing the importance of the individual

What is thinking? Vygotsky resists the separation of Thinking from Being. Thinking is a mode of acting in the world in the service of a human individual while acting in the world. It is NOT an autonomous stream of thoughts.

In Spinoza's philosophy the whole is more than the sum of the parts. Theorising begins with the whole, not the parts.

"a psychology that decomposes verbal thinking into its elements in an attempt to explain its characteristics will search in vain for the unity that is characteristic of the whole. When the whole is analysed into its elements, these characteristics evaporate because 'the internal relationships of the unified whole are replaced with external mechanical relationships between 2 heterogenoous processes'

To understand the parts begin with the whole.

Comment BK: NB. It doesn't follow that there is no place for decomposition, eg. phonics in learning to decode. But it does follow that there is a downside, you will miss out on holistic insights, if you ONLY focus on phonics

What other philosophies do is separate body and mind and then invent a way to put them back together

Descartes: pineal gland
Hegel: sublation (which is?)
modern: tools and signs as mediators

Vygotsky opposed to researchers splitting sound (sensible) from signification (supersensible)

Roth seems to argue that there was an evolution in Vygotsky's thinking from the whole is the sound-word (not the meaning of a word, that is suprasensible) to sense.

What is sense? It has visible and invisible aspects. Movement is part of the visible aspect. There is an inner sense which expresses the person's attitude to the goal, internal struggle, hesitation, motivation, hot temper, weakness, exaggeration, attainment for show. The movement may be in excess or less than what is necessary to achieve the goal.

Comment BK: I don't think he explains sense adequately

Comment BK: This is the clearest argument yet about why whole language advocates are against teaching stand alone phonics. Decomposition destroys the sensual activity of the whole.


Idealists make two mistakes:
1) Concepts over-ride practice
2) Blind to historical nature of concepts

Laws of dialectics, from Engels, and how Vygotsky interpreted them
a) law of transformation of quantity into quality. Vygotsky called these neoformations
b) law of interpenetration of opposites. Vygotsky followed Spinoza that mind and body are one substance which has a mind attribute and a body attribute
c) law of the negation of the negation. (?)

Research then and now makes the error of imposing concepts and principles from the outside. Method is decided in advance and concepts are imposed on whatever researchers are looking for. Methods and concepts should arise through a thorough investigation of the phenomena themselves, preconceived ideas and models. Data should create the theory, not the inverse.

Comment BK: Doesn't Roth do the same thing in the next section by proclaiming ethnomethodology as best?

You have to question your most cherished and your unconscious concepts

Vygotsky method: start with the most general "cell form" of the phenomenon and develop from there

On Methods for a 'Concrete' 'Human' Psychology of Mathematics

Benedict de Spinoza, The Ethics
Lucy Suchman, Human-Machine Reconfigurations: Plans and Situated Actions (2007), link goes to free pdf
Ludwig Feuerback, Spinoza (1833)
Evald V Ilenkov, Dialectical Logic: Essays on it History and Theory
Hegel Phenomenology of Spirit Part I (1807)
Lev Vygotsky Concrete Human Psychology (1989)
Lev Vygotsky Thinking and Speech
Wolff-Michael Roth Radical Uncertainty in Scientific Discovery Work (2009)
(I found a Roth paper online with a different title that seems to cover the same ground: Data Generation in the Discovery Sciences–Learning from the Practices in an Advanced Research Laboratory )
John Dewey and Arthur Bentley, Knowing and the Known (1999)
Wolff-Michael Roth and Alfredo Jornet, Situated Cognition (2013), link goes to free pdf
Georges Politzer Critique of the Foundations of Psychology (1928)
Bateson reference is missing from Roth's book (p.5)
Paul Ricoeur Time and Narrative
Bourdieu. The Paris Workshop

Ch 2 The Thinking Body

Ch 3 The Mathematics of Mathematics

Ch 4 Sociogenesis

Ch 5 Intra-Intersubjective Field

Speaker 1 and speaker 2 are having a conversation

Theory A: Independent speakers
The old way to look at it, perhaps associated with the early Vygotsky:
We have speaker 1 and speaker 2. The speakers are theorised as independent entities. Each speaker is caught up in their subjectivities so intersubjectivity becomes a problem. To solve the problem we invent sign mediation. The sign, eg. the word, has a physical body and an ideal meaning. It makes the connection between the material and the ideal world, the non physical or metaphysical world. The sign creates the capacity to bridge the gap between the 2 speakers. First we create a gulf between 2 independent speakers, then we create the sign to bridge the gulf we have created.

Theory B: Common, intersubjective speech field
A better way to look at it, Roth's interpretation of the late Vygotsky:
The speakers create with words a common or intersubjective speech field, a semantic field common to the interlocutors. The connection to consciousness, the struggle to achieve sense, occurs in that field. The sign, eg. the word, does not stand between two people but is the reality for two. The sign does not stand between the activity and 'meaning'. It is just a part of the making sense that the speakers embark on.

Constructivist viewpoint: Mind is constructed from interaction with materials out there in the world, what happens on the inside is explained through interaction with the outside. There are 2 distinct worlds, the mind inside, and the material world outside. The inner world, the mind, somehow creates reference to interpret the outer world.

Mathematical objects do not exist independently of people but are produced intersubjectively through dialogue and convention. Mathematics becomes a series of intersubjective social constructions.

In theory A the Self is isolated and intersubjectivity becomes a hard problem.

In theory B intersubjectivity is not a problem. It becomes a natural part of the joint labour that naturally occurs in tackling a problem.

eg. Consider the maths problem: How to divide 9 pies amongst 12 individuals? We can focus on the maths of the activity. There is no need to explain how the concepts that emerge in discussion get inside the minds to be retained by the students. Just do the maths, collaboratively.

Other theories:
Constructivist: discussed above
Enactivist / Autopoietic (Maturana): the cognizing organism is considered informationally closed
Behaviourist: the sign is not a mediating link in the behaviourist S-R model (Mikhailov reference, The "Other Within", not clear to me)

Historical-Philosophical Considerations

Human essence is societal relations. Societal relations are expressed in language, which is a tool of labour.

Language is our existing practical consciousness. It originates from our need to relate. Language is the key to consciousness, the historical realisation of Being.

Any tool of labour, like language, is inconceivable outside a common (intersubjective, not individual) field of purpose, sense, consent, sympathy and action.

How do we explain difference? Different thinking bodies have different movements and this produces different ideas.

The word is absolutely impossible for one person but possible for two (Vygotsky quoting Feuerbach)

When writing or reading private notes in a diary. A form of behaviour that initially involved another person is now directed by a person to the other in itself.

Even if there is only one person doing a proof there are two in proving. In proving the thinker divides into two: he contradicts himself and is only proven when it has held up to and overcome this juxtaposition.

What is the difference between Thought and Being?
Thought: Think you are hold $50 in your hand
Being: Actually hold $50 in your hand. You can feel it, see it, others can see it, it becomes societal, universal

Intersubjectivity is being with others.

We channel our thinking by modifying our material environment. In the old days, when we had handkerchiefs, you might organise to remember something by tying a knot in the handkerchief. More generally, speech is a material thing we have in common. (and today we have mobile phones)

The Transactional Field

Felix Mikhailov, The 'Other Within' for the Psychologist (2001)