The Philosophical Origins of Infinity
Before we examine the concept of the infinite more carefully, we should explore the problem of its origin. Perhaps in knowing where the concept of the infinite comes form we can learn something about its essential nature.
We usually form concepts either by reflecting on direct experience or by some form of mental abstraction involving some, particular, experience (either mental or sensible). What concepts actually are, or how they come to be utilized, is a difficult problem in itself. For our purposes here, we can merely say that concepts arise in us and that they somehow connect to experience. Our experience underlying conceptual thought is such that it can be categorized. In this way, the concept becomes a token or term representing a state or a thing. In most cases, that is, concepts are arrived at - or at least refined - though the use of language. For example, in forming the concept of a dog such that I can correctly use it to communicate or distinguish things called ‘dogs’, I must - at some point - have experienced a dog or dog-like entity. In order to understand what red is, or the taste of pineapple, I needed to have experienced cases of redness and the taste of pineapples. Once the experiences happen, I can abstract from the experiences themselves and remember them; subsequently I can then attach a term to distinguish one experience from another. This naturally brings up the question that concerns us here: what experience or set of experiences help us arrive at the concept of the infinite? How and from what do we abstract the concept of infinity?
If we can agree that we never immediately and directly or fully grasp the infinite in normal experience, i.e. in day to day life, then there are only two possibilities for explaining how the origin of the concept of the infinite originates.
(1) It might be that we intuit the infinite in expanded awareness , i.e. what some call 'mystical states', or
(2) perhaps we abstract the infinite from other experiences, e.g. the common experience of being aware of one finite thing/state and then being aware of another finite thing/state and at some point thereafter reflecting on how things exist. Upon reflection we can arrive at the subsequent intuition that whatever is determined exists as a thing and things are given to us with boundaries. These boundaries or limits can be transcended. If we reflect on the process of transcendence of established boundaries we arrive at a state beyond all boundaries: endlessness or boundlessness, and from there formulate a concept of the infinite and all the other qualities associated with infinity.
As an example of how the second approach to forming the concept of infinity might unfold, imagine reflecting on existence and the universe as a collection of things. Most people who do this eventually start to think about where the physical universe ends. This leads to the desire for answers to questions such as: 'how large is the world?', 'how large is our galaxy?', 'the cosmos?', '...the entire universe itself?' The possibility of infinity now arises. If mundane things exist within established boundaries, common sense tells us that there must be a context underlying the fixed boundary of any given thing and leading to a space or spaces extending beyond these visible boundaries. For simplicity's sake, we can label this extra-dimension ‘other than x’ (i.e. other than this thing, whatever ‘this’ is). We can also call this a transcendent property and if extended indefinitely a physical infinity.
While the extension of things in space and the potential endlessness of physical space itself, are obvious candidates for the origin of the concept of infinity they may not be the only ones. Are there other kinds of infinities? Here the limitless potential of thought itself and objects of thought come to mind. Whatever we can think about can be added to so that even the thought of everything can be said to be able to be apprehended in reflection. In this way we can come to think about everything plus one. Mathematics offers itself as the natural realm to seek the expression of this second, abstract, kind of infinity. In fact, if anything at all is abstract, many would agree, the complex objects forming the theoretical basis of the mathematical disciplines fit that bill. So we have a second kind of infinity: mathematical infinity arrived at through abstract thoughts about objects in our minds. A unit and a unit make two units, but the upward number of potential units seems to be extendable to an endless degree. Moreover, as we'll see, physical and mathematical infinities are both also often separated from the mystical intuition of the all or absolute that is discussed by some philosophers and theologians. Some thinkers have called this absolute infinity or the existence of a highest being infinite, and held that God is infinite. If God exists, he would not be a number or a physical object or thing. Therefore we have arrived at a metaphysical concept of infinity. Therefore, before proceeding let’s take a look at the most common characteristics historically ascribed to the concept of the infinite. Here the through philosophical study of infinity by the British philosopher A. W. Moore, provides a helpful starting point.
According to Moore, the label of “the Infinite” has been understood and applied to all of the following concepts:
- A) Boundlessness, Endlessness; Unlimitedness; Immeasurability; and Eternity, on the one hand. Also to,
- B) Perfection; Completeness; Wholeness; Absolute Unity; Universality; Absoluteness; Self-Sufficiency; Autonomy, on the other.
As A. W. Moore points out in his book The Infinite, “The concepts in [A] are more negative and convey a sense of potentiality. They are the concepts that might be expected to inform a more mathematical or logical discussion of the infinite. The concepts in the second cluster [B] are more positive and convey a sense of actuality. They are the concepts that might be expected to inform a more metaphysical or theological discussion of the infinite” (Moore 1990:2)
So we find the concept of the infinite overlapping across the realms of: logic, mathematics, physics, metaphysics (i.e. what, if anything, is beyond the physical) and theology. For the sake of completeness we should also add aesthetics, since the infinite can find expression in artworks.
One of the main goals of this website is to catalogue, examine, explore and analyze the many different expressions of the infinite as used across different disciplines spanning both the sciences and the arts.
Philosophical orientation of the current Project
Philosophy is, amongst other things, the intellectual discipline that studies argumentation and seeks the truth about the world and our place in it. Unfortunately academic philosophy is currently in a state of methodological chaos. Modern philosophy cannot decide on whether it wants to be an accessory to the natural sciences (something it isn't suited for), a simple history of ideas (we already have historians), quasi-poetical revelry or economic-political critique (footnotes to Marx or Heidegger), or a kind of analysis of language and study of our feelings. This situation is depressing. But there's another tradition, extending outside of professional philosophy and more closely connected to philosophy's original mission. This is the Socratic idea of philosophy as the search for truth and the analysis of our beliefs. It is this approach, informed by metaphysical thinkers such as Plato, Aristotle, Descartes, Kant and Leibniz, that will inform the framework developed herein to analyze infinity. Viewed in the above way, moreover, it can be argued that philosophy is the ideal discipline to use in approaching the study of basic or essential concepts since only a genuinely philosophical approach can connect different disciplines together in order to arrive at a broader perspective and clarify our understanding of the world and ourselves in a fundamental way. Philosophers in the above tradition have often relied on two tools in order to formulate theories and arrive at a deeper understanding of the world. The first of these tools is logic which must be made use of to analyze concepts and scrutinize the structure of reasoning involved in the search for truth, and the second is the use of insight and imagination. By insight and imagination is meant the careful exploration of the world and the analysis of possibility and necessity in a broader sense than would be done by empirical scientific account of facts. Science. although essential to discovering facts about about the world and testing theories is not philosophy and presupposes a metaphysics. By using logic, by contrast, is meant analysis making use of both formal and informal logical tools. Logic is (formally) the study of the structure of thought and the exploration of the coherence and rational properties of reasoned arguments. Logic is also a tool that can allow us to clarify, apply, organize (and occasionally) increase our knowledge. Finally, logic can be said to have truth as its normative goal. Since logic aims at elucidating what makes something true, it can be seen that logic is essential for science. However, even so, not all sciences are strictly formal. Inductive inference and experiment also make use of assumptions and methods that are informal but essential to modern science.
In a formal sense, applying logic in its pure or theoretical form, all arguments are either: valid or invalid (i.e. they either contain or avoid contradictions). For the purposes of our study we can call an argument a 'well formed theory' or set of assertions that aim to determine a truth-value. All arguments have premises and since we can argue about or for and against almost everything, an essential part of studying and evaluating arguments (such as whether anything is actually infinite) is by examining our assumptions and what can called the premises of our arguments and assessing whether those premises are either sound (true and provable) or unsound (untrue and not provable). When studying soundness, in other words, we must move beyond merely formal concerns. If I argue, for example, that the world is really made of cheese, I can formulate a valid argument based on this premise:
- Hypothesis: (1) The world (and the totality of things in the physical universe) is actually cheese
- Premise: (2) John is a part of the world, therefore
- Conclusion: (3) John is made out of cheese.
From the hypothesis that everything is made out of cheese, and the premise that John is a part of the world, I deduce that John is made out of cheese. This is a valid argument (in fact it’s a valid syllogism, i.e., an argument having three steps and deductively arriving at its conclusion through a middle premise or term). A valid argument is any argument whose conclusion is a logically derived from its premises. So, the above argument is valid, but it is unsound. It is unsound because the world is not made out of cheese. Cheese is a product and part of the world but is itself made up of other more essential properties (atoms, particles, etc.). So validity and non-validity are different from soundness and unsound assumptions. The analysis of form is important, however, because no invalid argument can reliably be said to be true and the presence of a contradiction is the sure-fire sign that an argument is suspicious.
An invalid argument rarely leads to deeper understanding or truth:
- Hypothesis: (1) All Human beings are mammals,
- Premise (2) Socrates is a Human being,
- Conclusion (3) Socrates loves Cheese.
This is a very different argument from the valid one given above. Notice that the hypothesis of this argument, so far as we know, is also sound. But the form is invalid and consequently it is not a very useful argument to help us arrive at any deeper understanding of the ideas it employs. Any strictly logical study of the infinite runs us into problems because of the complex nature of the infinite as a concept. We really don’t know if the hypothesis holding that the infinite actually exists is sound or unsound. Arguments can be made and offered for both cases. Therefore, for reasons of clarity and intellectual cleanliness, we can separate the logical and mathematical from the physical and metaphysical/theological senses of the infinite. What grounds can we give for separating the mathematical (formal) from the physical (non-formal) senses of the infinite? In the course of our investigation the reasons for separating the infinite will become more clear, for now, we can say that problems emerge when the two senses are united.
Aim of these webpages
In what follows we will examine the concept of infinity and what it might mean. In the process we will examine some of the most important ideas and instances of the application of the concept of infinity. The infinite has influenced our culture and its impact has been felt in religion, art, science and philosophy. Because the concept of infinity is so broad and complex the basic approach taken by the course will be broadly philosophical but with a special emphasis on the scientific or quantitative (physical and mathematical) applications of the infinite.
The methodological approach we take to analyze the infinite will also be chronological. The course is divided into four sections:
- Infinity in the ancient world,
- Medieval conceptions of the infinite
- Renaissance [Early Modern] conceptions of the infinite
- The Infinite in modernity – The science of the transfinite
- The Infinite: a philosophical theory about its true nature
Each main section will, in turn, explore how the concept of infinity has been examined historically and how different cultures and traditions have reacted to the infinite. The possible existence and potential meaning of the infinite as it relates to both the world and us is the primary focus of this study.