On Abstraction and Reality

1.1 On Systems of Varying Natures:

The start of this essay could well be considered a kind of thought experiment. I intend to try and define two different ideas, compare them and use a unique property of one to shed light on the other. There is no particularly significant philosophical interest in the first of these concepts that I am to consider, but I will try and use it as a way of illustrating features – or lack thereof – of what are hopefully more noteworthy concepts. This section may seem excessive and unnecessary in relation to the point that I am trying to make, which perhaps it is, but nevertheless I intend to include it, as it is the way in which I understand it, and so the way in which I am most comfortable attempting to convey it, and in which that I believe will give the strongest basis for progression. I will be discussing systems of varying natures, namely those systems that are abstract and those that are physical, specifically focusing on the abstract mathematical system and the system of physical reality in which we inhabit. And so, to begin, I must first clarify what it is that I mean by an abstract system and a physical system – not at all a trivial task, for reasons that will become more apparent as I try to construct a definition.

 

Attempting to find an approach – or distinction – that allows an objective definition of an abstract system and of a physical system is not straightforward. Whilst the concepts that I am trying to convey have differences, it is unclear as to which of these is the truly defining one. I could label a physical system as one with which the body can interact, and an abstract system as one with which the mind can interact. This has the major drawback of potentially devolving into – or being misunderstood as – a variant of Platonic dualism, and thus implying a distinctness between the realms of mind and body. To bypass this potential confusion, I could instead attempt to define the distinction to be ‘that which exists solely in the mind’ and ‘that which exists solely outside of the mind’ but then it would then be necessary to define existence which is outside of the scope of what I currently intend to discuss. Also, in this case, the ambiguity of whether ideas such as colour exist independent of the mind or require perception in order to exist – so depend on the mind – will emerge, and it is these pedantries that will detract from what it is that I am trying to say.

Alternatively, I could appeal to intuition, explaining roughly what I mean and giving examples rather than attempting to nail down an objective definition:

“An example of an abstract system is the logical system of mathematics, which is a non-tangible, theoretical system, where the objects in the system themselves do not exist physically.”

“An example of a physical system is the external world that we inhabit, a tangible system, where the objects in the system themselves exist.”

But this is a suitably vague way to begin an essay, as this is a loose definition, which, although conveys meaning to me – as it my definition and so I am required to understand what it is that I am defining in order to have defined it – it may not convey the same meaning to those that do not understand the distinction already. It ceases from being objective, and the words obscure the meaning and introduce an element of subjectivity, which is exactly what it is that I am trying to avoid. It also relies on the fact that the concept itself is so obvious and so well embedded in common thought, that all that is needed to evoke it is a single example and indication as to the central part of said example– which in this case probably will be a misassumption.

Now perhaps it is that it is my fault that I cannot define it. Perhaps I have already made an error in the naming of the concept, by labelling it a ‘system’, and so there is no way I can recover a definition which conveys what it is that I understand. I have used the word system because my idea of a system equates the closest to the idea that I wish to express than any other word and associated idea, but in actuality, the word ‘system’ is just a placeholder, a mere name and it need not inform the reader of anything regarding what it describes. Consequently, I could have chosen any arbitrary name or concocted my own word to label it, but I have not, so the placeholder word ‘system’ becomes something that is used to try and evoke what it is that others would understand a system to be, and hence via analogy understand. The placeholder hence becomes subtly part of the definition or at least a way of narrowing down the scope of what it is I am discussing, but is not sufficient to give an unambiguous definition by itself, since it would be reliant on a univocal understanding of the word ‘system’ which is not the case. The placeholder is used to evoke an analogical response, but again the subjectivity presents a limitation to the method.

Or perhaps it is simply not definable in terms of words. Perhaps it is necessary to understand the theoretical concept before any attempt to understand the definition of it in words can be made. In this case there is no way to transfer the idea from one person to the next. As such the inefficiencies and very nature of language become a barrier to communication and the idea is inexpressible and so not discussible as a universally understood concept.

 

The problem that I am having here is one of the problems that I intend to discuss, and so I will use it as a point of reference as I continue. It is somewhat convenient that this essay begins with an example of the point that I am trying to make, although also detrimental to the expression of said point, as this initial vagueness will undermine and convolute its meaning. ­Plainly, there is no way for me to define what it is that I wish to talk about in such a way that everyone interprets it identically. As such, I will have to attempt to amalgamate several of the aforementioned different methods to try and minimise ambiguity in the way I define it. To begin with I am only really going to be focusing on and comparing the logical system of mathematics and the system of physical reality that we exist in – for now at least – so I need only really define what it is that I mean in relation to these two systems.

 

Now by ‘the system of mathematics’, what I mean is everything that exists as an abstract, intangible concept in mathematics, and the logical methods that connect them all. I do not mean simply those things that are mathematical in nature but exist in reality and are tangible. For example, consider the difference between an idealised conceptual triangle in mathematics, and an imperfect triangle in physical reality. Now a mathematical triangle has perfectly straight edges, and angles that sum to exactly 180 degrees (on a Euclidean Plane at least), whereas a triangle in physical reality lacks these properties.[1] Even if one were to draw out a triangle in a mathematical context it would not be the same as the idealised form, due to the imperfections in the way it was drawn, or the implements used. This is also due to the fact that the parts a triangle is composed of are not physically existent. In his Elements, Euclid defined a point to be;

‘that which has no part’

 and a line to be;

‘a length without breadth’

Clearly neither of these two concepts can physically exist as they both rely on existence fused with an absence of a physical element – an absence that nullifies tangibility. A triangle is composed of lines meeting at points in a specific way, so it follows that there is no way that a mathematical triangle can truly physically exist, and as such can only be conceptual inside the system of mathematics.[2] So an attempt at emulating an abstract object in physical reality is doomed to failure, as it loses the properties that define it, and so the emulation becomes nothing more than an approximation – or visualisation – of the absolute object it represents. The members of the system of mathematics are the mathematical objects, not the physical versions of these objects.

Similarly, the ‘system of mathematics’ is connected via logical reasoning and rules, used to move from one statement to another and provide properties to abstract mathematical objects, starting from a set of consistent axioms. These are what link mathematical structures and statements together and hence they give the whole system interconnectedness, and ensure that it is rigorous, however these links are not merely ways of describing the associations in the system but also a part of it themselves. This system is also well-defined since, in mathematics, given identical starting definitions and axioms, and obeying the rules of logic, there cannot be any difference in conclusion regarding an object or statement in the system.[3] These axioms and reasoning are themselves abstract and intangible, and furthermore the entirety of the system of mathematics exists conceptually and independent of the physical world – not in that the property of existence allows us to interact and change things using purely our minds, but rather that basic definitions bring objects into an abstract existence, which necessitate the existence of truths regarding said objects, more complex structures and so the system itself.

 

The ‘system of physical reality’ is harder to explain, since what I mean by ‘physical system’ is further from the standard understanding of the word ‘system’ than what I mean by an ‘abstract system’[4]. In fact, the word ‘system’ is probably not the best word I could have used – it is unnecessary even – and perhaps convolutes the term ‘physical reality’. The reason I have used it is to illustrate an element of closure on objects in this reality, so there is a clear distinction between that which is physical and that which is abstract. But also, since I have used it in the names of both physical and abstract, I am attempting to show that there is an element of similarity between the systems themselves. In essence it is used to show that the contents of each system are distinct, but the structure of the systems themselves share similarities.

Now what I mean by the system of physical reality encompasses all that our physical bodies can interact with on a daily basis. I sit on a seat at a table, typing on a laptop, drinking coffee. I physically interact with all of these things. Someone sits next to me typing too, I can speak and interact with them. I can get up and move to a different physical table and sit down on a different physical seat. I can order more coffee and drink that. I can get up and leave. In fact, everything that I can perceive and interact with is a part of the physical system, except the thoughts that exist as non-physical entities in my head before I write them down.[5] My body exists in the physical system, and likewise my brain does too, but the thoughts that I have are intangible, and are not physical by definition and hence are not a part of the system.

 

So, the mathematical system and system of physical reality are examples of abstract and physical systems respectively. Both can essentially be tautologically categorised by the fact that everything that is in an abstract system is abstract and intangible and everything in a physical system is physical and tangible. Now one may consider the last two thousand words superfluous to requirements, and argue that it would have been better for me to just define them as I did in the last sentence, but the reason I have not – and instead have given a more rigorous definition of two specific examples – is to attempt to limit the potential ambiguity and pedantries that may have arisen from being too vague. It is better to be both figuratively and literally on the same page as I continue, as then confusion will be minimised. Whilst I hope this is the case, many may find the definitions unclear, and consequently I hope that as I continue to discuss these concepts, the words that surround and compliment them and the context within which they are used, begin to shed some clarity on what it is that I mean by these two types of system.

[1] Triangles can have curved edges, and angles that sum to more than 180 degrees on, for example, a sphere. This is a pedantry which is irrelevant to the point that I am trying to make.

[2] Whilst this seems to share multiple connections with the ‘Platonic Forms’ ideal, I would like to clarify that I am not extrapolating as Plato did, and postulating the existence of some other realm, where all these forms are contained.

[3]However, by Gödel’s incompleteness theorem, given a logical system with consistent axioms, this system can never be complete.

[4] Or at least further from my perception of the standard understanding.

[5] Brain activity is different to the thoughts themselves. Whilst the brain activity is physical it is not the same as thoughts, it is instead a mere indication that thoughts are occurring. It is possible that we may one day be able to translate from brain activity to thoughts, but the fact it would be a translation implies qualities of bijection and isomorphism, but not equivalence.