Just as there is an ‘Aboriginal Physics’, there is a corresponding ‘Aboriginal Mathematics’ and the purpose of this page is to provide a narrative description of how the mathematics of Aboriginal Physics differs from the mathematics of Enlightenment physics, following the distinctions on ‘mental tendencies of mathematicians’ made by Henri Poincaré in his “Dernieres Pensées’.
Aboriginal/medieval man, seeing himself as included in and defined by a ‘web-of-life’, ‘accounted for’ or ‘counted for’ things in a different way than our Western culture a-counts for them (vis. everything seems to go back to ‘number theory’ in the mathematical view). For example, if we take the Aboriginal view that ‘the farmer is the result of the production of grain, not the cause’ (corresponding to the view that ‘ man is merely a strand in the web of life, he did not invent the web’), … the different spatial-relations that associate with this arrangement give rise to a different way of ‘ a-counting’.
Pasteur’s example, wherein ‘the proliferation of bacteria is the result of illness (the terrain-dynamics) rather than the cause of them, may be easier to assimilate than ‘the proliferation of farmers is the result of terrain-dynamics, rather than the cause of them’.
In the Enlightenment view, the local causal agent ‘is equipped by us with’ his local causal powers as we visualize his local dynamics relative to an absolute fixed and empty (Euclidian) reference space. This absolute space framing is what allows us to ‘see’ his actions as if they were ‘his own’.
[N.B. This 'idealising' of him as a 'local causal agent' with his own locally originating behaviour breaks down if he is engulfed in a lava flow or a tsunami or carried off in a tornado since the mind can no longer deny that the 'real reference frame' is the terrain-dynamic that he is included in].
We do this ‘notional equipping of representations of things with their own local causal agency’ for storm-cells in the atmosphere, which gives us the (false) understanding that the storm-cells have their own locally originating behaviour which gives us, in turn, the (false) impression that ‘they are stirring up the fluid-dynamics of the atmosphere’. In fact, it is the fluid dynamics or ‘flow’ of the atmosphere that not only create the storm-cell but shapes its behaviour. Again, the Enlightenment ‘seeing’ approach is to ‘frame’, in a viewing field, the storm-cell and give representation to its dynamical form in terms of a ‘local image’ as in a photograph. Now that we have removed the ‘parentage’ of the flow (that creates the dynamical form and shapes its behaviour) and re-instituted the parenting (creative) powers in its local representation, we are forced to invoke the notion of ‘time’ in order to ‘animate’ the representation’. Thus we take a series of representations at ‘different times’ and by re-presenting them in sequence, as in frames in a film, we are able to ‘animate’ the now ‘apparently local’ form ‘in space and time’. Enlightenment physics does this ‘removing of flow-based parentage’ and ‘re-instituting it in a notional ‘local object/organism-self’, not just for storm-cells, but with all manner of dynamic forms, from storm-cells to organisms and human beings. The ‘systems sciences’ were in fact ‘born’ in the 1950s for the very purpose of shaking up this ‘closed systems’ view propagated by Enlightenment physics.
Mathematically, this gives rise to two different ways of perceiving ‘objects’ which Poincaré refers to as ‘Cantorian realism’ and ‘pragmatist idealism’ (the latter is his way of perceiving and also the way of Aboriginal physics). “The scientists of the two schools have opposite mental tendencies.”
For a full discussion of this ‘split’ in the ranks of mathematicians on the basis of how the view ‘objects’ and how they infuse their view into their mathematics, see ‘Poincaré – Dernières pensées , Chapitre V : Les Mathematiques et la Logique’.
A question that discriminates between the two views is; ‘Do objects exist before we observe them?’ The Cantorian realists will answer ‘yes’ while the pragmatist idealists will answer ‘no’.
Pragmatist idealists, according to Poincaré; “…believe that an object exists only when it is conceived by the mind and that an object could not be conceived independently of a being capable of thinking. There is indeed idealism in that. And since a rational subject is a man, or something which resembles man, and consequently is a finite being, infinity can have no other meaning than the possibility of creating as many finite objects as we wish.”
“But the Cantorians are realists even where mathematical entities are concerned. These entities seem to them to have an independent existence; the geometer does not create them, he discovers them. These objects exist, so to speak, without existing, since they can be reduced to pure essences. But since, by nature, these objects are infinite in number, the partisans of mathematical realism are much more infinitist than the idealists. Infinity to them is no longer a becoming since it exists before the mind that discovers it. Whether they admit it or deny it, they must therefore believe in actual infinity.”
This ‘split’ amongst mathematicians that impacts on logic and the potential for contradictions therein, corresponds to the split between the Aboriginal/Medieval ‘way of seeing’ and the Enlightenment representation ‘way of seeing’. That is, in a flow-continuum involving the continual gathering and re-gathering of dynamical forms (imagine ‘boils’ in a pot of boiling water or ‘storm-cells’ in the atmosphere), the ‘greater reality’ is the dynamical flow-continuum. As observers, we can capture ‘representations’ of the dynamical forms that emerge in the flow, but our lives are not long enough to observe an infinity of them. However, our representations of dynamical forms, if we take them for ‘reliable substitutes for the visible’ as occurred in Western Enlightenment, are not limited in number. If we believe that our representations of storms are ‘reliable substitutes for the visible’ we then impute to the originally experienced/observed phenomenal forms, the ability to exist before any one observes them, and this further implies that there is an infinite supply available to us; i.e.;
“[Cantorian realists] … think that objects exist in a sort of large store, independently of any mankind or of any divinity that could talk or think about them; that in this store we can choose freely; that no doubt we do not have a good enough appetite nor enough money to buy everything; but that the inventory of the store is independent of the resources of the buyers. And from this initial misunderstanding [relative to the pragmatist idealists] all sorts of divergences in detail result.”
“The debate could go on for a long time; but the point which I would like to emphasize is that if this type of definition were permitted, logic would no longer be sterile, and the proof is that a multitude of arguments have been formulated in this manner which are destined to prove propositions which were in no way tautologies since there are persons who wonder if they are not false. Therefore, we are amazed at the power which a word can possess. Here is an object from which nothing could be derived until it had been christened; all it needed was to be given and name and it worked wonders. How can this happen? It is because, by giving it a name we have asserted implicitly that the object did exist (that is, was free from all contradiction) and that it was entirely determined. But we do not know this at all according to the pragmatists.”
It is not hard to draw parallels and in fact ‘to equivalence’ this split between mathematical ‘mental tendency’ to the split between the Aboriginal/Medieval and Enlightenment ‘way of seeing’. And, as Poincaré observes, the instrument for bringing mathematicians into agreement is ‘the proof’, however, in this case, since the proof deals with two different concepts of infinity and since our lives are finite, there will not be any proofs to establish which view is correct and which view is false.
So, mathematics and logic are not where we can look to resolution of the two ways of seeing; i.e;
“At all times, there have been opposite tendencies in philosophy and it does not seem as if these tendencies are on the verge of being reconciled. It is not doubt because there are different souls and that we cannot change anything in these souls. Therefore, there is no hope of seeing harmony established between the pragmatists and Cantorians. Men do not agree because they do not speak the same language, and there are languages that cannot be learned.”
‘Aboriginal physics’ implies the mathematical view of ‘pragmatist idealism’, wherein we accept that ‘representations’ of dynamical forms are ‘idealisation’s that must not be confused for reality. The ‘storm-cell in the flow’ view of objects is consistent with relativity and with the conjugate habitat-inhabitant relation (conjugate space-matter relation) of Ernst Mach; “the dynamics of the habitat condition the dynamics of the inhabitants at the same time as the dynamics of the inhabitants are conditioning the dynamics of the habitat.”
In this ‘Aboriginal physics view’, as with the storm-cells in the flow of the atmosphere, the ‘inhabitants’ and their dynamics are the ‘result’ of the habitat-dynamics rather than their ‘cause’. In the dynamical habitat of the universe, the dynamical inhabitants we call ‘planets’ gathered, and in the dynamical habitat of the planet ‘earth’, the dynamical inhabitants we call ‘plants’ gathered, and in the plant-growing habitat, the dynamical inhabitants we call the farmers gathered. Thus, farmers are the result of plant-growing, not the cause of it.
This ‘parenting precedence’ is qualitative and spatial (extensional) as in the ‘pragmatist idealist’ mathematical mental tendency. If we base our way of seeing on ‘representations’ where the parenting power is shifted to the inside of the local object-representation, the sense or something being causally produces is, on the other hand, seen as a function of ‘time’ and ‘quantity’ ‘radiating outwards’ from the ‘local causal agency’ implied by the ‘representation’.
There are thus two different ‘mathematics-and-logic’ ‘mental tendencies’’ that respectively back up ‘two different ways of seeing’.
In order to quiz yourself on which mental tendency you are putting into precedence, ask yourself whether you believe that storms cause the turbulent flow in the atmosphere or whether the storms are the result of turbulent flow in the atmosphere. And if you ‘get by’ this question on the side of Aboriginal physics (pragmatist idealism); i.e. that storms are NOT local causal agencies as we tend to represent them (for ease and convenience) in meteorological reports, but are spawned by changes in the flow of the atmosphere (e.g. by solar irradiance bringing the circulating ocean currents and fluid atmosphere ‘towards the boil’), then you can follow up by asking yourself whether bacteria and viruses cause the unbalanced dynamics in the body that we call ‘illness’ or whether bacteria and viruses are the result of the unbalanced dynamics in the body (as Pasteur and Béchamp contended). If you came out on the side of Pasteur (which puts you crosswise with the modern medical establishment), you can follow up with a question as to whether the storm-cells we call criminal-attacks, terrorist attacks etc. are the cause of disturbed dynamics in the social terrain or the result of them.
It would seem clear that the above-described split in mental tendencies in mathematics and logic is sure to resurface as a split in mental tendency in our social and political way of seeing.
This leaves us with alternative ‘precedences’ of response to destabilisation in the dynamics of habitat, as is apparent in medicine; (a) cultivate a restoring of balance in the terrain (Pasteur, Béchamp, Aboriginal Physics, pragmatist idealism, ‘holistic’ medicine) or (b) eliminate local causal agents that are creating turbulence in the terrain-dynamic (modern medical establishment, Enlightenment Physics, Cantorian realism)
* * *