December 2023
What is imagination?
Please reference as: Maria Tamboukou. 2023. 'What is imagination?', https://sites.google.com/view/numbersandnarratives/newsletter/december-2023
My Dear Mrs Somerville,
Can you tell me if any solid models have ever been made for illustrating some of the Propositions of Spherical Geometry, and if so where such things are best to be had. Next to this, some extremely good plates on the subject would be a great help. The kind of propositions I refer to are those on the intersections of Circles of the Sphere; for instance the following, which I take from Spherical Geometry which precedes Lardner’s Spherical Trigonometry [...] These are enough to put me in despair and I have been in danger of turning crazy in trying to imagine the circles in my mind’s eye.[i]
In this letter above, written from her residence at Ockham Park on 25 March 1836, less than two months before Ada Lovelace gave birth to her first child, Lovelace expresses her need for visual and tactile representations of mathematical concepts. In response, Somerville promised to ask for Charles Babbage’s help, but also tried to reassure Lovelace’s anxiety:
Pray don’t let the circles turn you crazy till we meet, for I am sure I can explain them to your satisfaction viva voce, though I doubt of my talents that way on paper’. [ii]
This idea of ground mathematical abstractions were not however always seen positively in the nineteenth century world of science. As a matter of fact, women’s aptitude in mathematics was being dismissed precisely on the grounds of their supposed incompetence for abstract thought, even amongst the philosophers of the Enlightenment. Immanuel Kant for example had famously argued that women’s mind was different from men, in that it was not made for abstract thinking, concluding that ‘a woman therefore will learn no geometry.’ (1960, 79) Lovelace seems indifferent to such ideas and appeals to Somerville for help in imagining the circles of the sphere. Coming from an older generation, as she was thirty-five years older than Lovelace, Somerville seems to have internalized such discourses as a famous extract from the second draft of her Recollections, which was never published, reveals:
In the climax of my great success, the approbation of some of the first scientific men of the age and of the public in general I was highly gratified, but much less elated than might have been expected, for although I had recorded in a clear point of view some of the most refined and difficult analytical processes and astronomical discoveries, I was conscious that I had never made a discovery myself, that I had no originality. I have perseverance and intelligence but no genius, that spark from heaven is not granted to the sex, we are of the earth, earthy, whether higher powers may be alotted to us in another state of existence God knows, original genius, in science at least, is hopeless in this. (Somerville 2001, 168)
Somerville follows here what Ruth Messbarger has identified as ‘the double-voiced discourse that simultaneously defied and affirmed misogynist constructions of femininity’ (2005, 18) in the early modern period and beyond. On the one hand she defends women’s right to education and therefore engagement with science, but on the other hand she offers an apology for the defects of her sex, by acknowledging her position as an expositor rather than creator of scientific knowledge, a creature ‘of the earth’ and not of the sublime spirit. Unfortunately this discourse that there were not really great women mathematicians has reached our days, despite the epistemological critiques of what counts as great in the history of science and mathematics and more widely what is understood as a contribution to scientific progress (Schiebinger 1991, Fara 2004).
Lovelace’s intervention in such discourses was crucial: at the same time of admitting that she was ‘turning crazy in trying to imagine the circles in my mind’s eye’ she also held the firm belief that imagination should delve into the unexplored by ‘seizing the unseen’. Her 1841 essay on the importance of imagination in the mathematical sciences prefigures a burgeoning body of literature around the mathematical imagination, as well as the importance of visual thinking in mathematical practice. (see Mancosu 2008) In Lovelace's’s configuration, imagination is twofold: it combines, ‘bringing together things, facts, ideas, conceptions, in new, original, endless, ever varying, Combinations’[iii] and ‘it conceives and brings into mental presence that which is far away, or invisible, or which in short does not exist within our physical and conscious cognizance.’[iv] In this light imagination is ‘the Discovering faculty […] that which penetrates into the unseen worlds around us, the worlds of Science.’[v] Thus Lovelace needs ‘geometric models’, as material aids for the work of imagination to unfold: ‘Mathematical Science shows what is. It is the language of unseen relations between things. But to use and apply that language we must be able fully to appreciate, to feel, to seize, the unseen, the unconscious.’[vi]
Lovelace’s take on imagination as a ‘discovery faculty’, a path to the yet ‘unseen’ echoes Sophie Germain’s philosophical approach to the creative forces of imagination. Being the first woman to win the Grand Prix des Mathématiques from the French Institute in 1816, one year after Lovelace’s birth, Germain was also an important philosopher, whose work had been celebrated by her contemporary philosophers, such as Auguste Comte.
In her philosophical work, Germain compares the impressions we get from fictional and scientific works and concludes that there are no important differences between them. In making these comparisons, she carefully demonstrates the identity of intellectual processes both in poetry and in science by showing that there is a continuous interchange of feelings [sentiments], imagination and rational reasoning in the way they unfold. For the poet there is ‘a tumultuous struggle’ of abstract images and opposing projects until a simple idea finally emerges. (Germain 1896, 82) For the mathematician there is also a simple, ‘fruitful idea’ that arises through [his] struggle with imagining a new problem in areas already researched and established.
There is an interesting genealogy in the way women mathematicians, scientists and philosophers wrote about imagination in the eighteenth and nineteenth centuries that this project is in the process of excavating.
[i] AB to MS, letter dated 25 March, 1836 [Oxford Bodleian Libraries/Mary Somerville Collection/c.367, f.70].
[ii] MS to AB, n.d. [Oxford Bodleian Libraries/Archive of the Noel Byron and Lovelace Families/Dep. Lovelace Byron/174].
[iii] Essay on Imagination, 5 January 1841 [Oxford Bodleian Libraries/Archive of the Noel Byron and Lovelace Families/Dep. Lovelace Byron/175].
[iv] Ibid.
[v] Ibid.
[vi] Ibid.
References
Fara, Patricia (2004). Pandora’s Breeches. London: PimlicoGermain, Sophie (1896 [1879]) Œuvres philosophiques de Sophie Germain, suivies de pensées et de lettres inédites. Et précédées d’une notice sur sa vie et ses œuvres par Hte Stupuy (Nouvelle Édition). Paris: Firmin-Didot.Kant, Immanuel. 1960 [1764]. Observations on the Feelings of the Beautiful and the Sublime, translated and edited by John T. Goldthwait. Berkeley and Los Angeles: University of California Press.Mancosu, Paolo, ed. 2008. The Philosophy of Mathematics. Oxford: Oxford University Press.Messbarger, Rebecca. 2005. “The Italian Enlightenment Reform of the Querelles des Femmes”, in The Contest for Knowledge, edited and translated by Rebecca Messbarger and Paula Findlen, 1-22. Chicago and London: The University of Chicago Press Schiebinger, Londa (1991). The Mind Has No Sex? Women in the Origins of Modern Science. Cambridge MA: Harvard University Press.Somerville, Mary. 2001 [1873] Queen of Science: Personal Recollections of Mary Somerville, edited and introduced by Dorothy Mcmillan. Edinburgh and London: Canongate.