This is an excerpt from the book series Philosophy for Heroes: Knowledge. You can get a copy here.
To those who do not know mathematics, it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature. […] If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in.
—Richard Feynman, Character of Physical Law
Why did humankind develop a system of mathematics?
The origin of Mathematics can be found most likely well before that of written language. With the centralization of communities (such as in the Sumerian kingdom, around 2300 BC), and with job specialization, increasing population density, and stationary settlements, features of society such as calendar systems, astronomy, trade, land surveying, work-sharing, storage of goods, and currency gained importance. The more accurately a king knew the extent of his land holdings, the more accurately he was able to tax them—just as farmers could cultivate their fields more productively when they knew exactly what calendar day it was, or as astronomers could better predict the movements of bodies in space when they learned concepts of geometry. These applications revolved particularly around enumeration and measurements. Also, theories related to irrational numbers already existed in Greek antiquity (ca. 500 BC), dealing explicitly with lengths of diagonals and implicitly with the application of the “golden ratio” in the arts, but a scientifically sound definition of irrational numbers, as well as the field of set theory, did not exist until after the Renaissance, the heyday of mathematics.
Mathematics arose from the need to count quantities, to compare quantities, and to describe processes.
In the following, we will focus less on general mathematical history than on the coupling of language and the representation of entities. In particular focus, we will consider the question of what a “set” is, what a “number” is, and how and in what form we should use those concepts. The reason for this chapter is that in discussions, aspects of mathematics are often used wrongly as an ontological or epistemological argument. A clear understanding of what mathematics is, in relation to the philosophy that we have discussed so far, can help prevent or resolve misunderstandings.
Sets are not, themselves, entities; what are they?
Sets are a central subject both in mathematics and in daily communication. The question is: What are sets? Do they exist? Do they have the properties of their elements? Is there an invisible tape that connects all elements? Let us first take a look at the different ways the term “set” is used:
- An enumeration of definite concepts, entities, or quantities of definite entities (“apples, pears, oranges,” “these apples,” “these three piles of apples”)
- All entities belonging to a concept (“all apples”)
- All entities defined by a recursion (“the natural numbers,” “the rational numbers”)
- All entities specifiable by an infinite generative process (“the irrational numbers”)
- All self-referential sets (“the set of all sets”)
According to our definitions of “entity” and “existence,” sets are not entities since they do not have properties with which they could interact with entities. If you place five oranges together, you do not suddenly end up with more than five oranges. An (unstructured) set never contains more members than those determined by the properties that define the set. The properties of a set are strictly mental constructs that describe its members.
Basically, sets correspond to concepts. The difference is that concepts are not defined as enumerations of entities, but rather as the enumeration of properties of entities. For example, the concept “chair” could be defined by the properties “chair legs,” “seating,” and “optionally a backrest,” while we enumerate a set of three chairs for example by “this blue chair,” “that green chair,” and “that big chair over there.” A given set could consist of only a portion of all existing entities of a concept (e.g., of all existing elements of a concept except for one), while concepts always refer to all existing associated entities (and entities that will be discovered in the future or that existed in the past).
For a better understanding, let us consider another set, namely “all people residing in New York.” What exactly is this set? Would it be meaningfully defined or comprehensible if everyone were tied together with string? Or does this set correspond literally to the written sentence, the ink on the paper?
It should be apparent that we need more in order to be able to intelligibly grasp and define this term. If we consider the set of New York residents, then we do not need to assemble millions of people in an office in order to deal with them. Rather, we deal with pointers, such as a list of telephone numbers or addresses. Thus, we could describe the set of all New Yorkers for example in the form of a thick telephone book. As the sentence itself asserts, the telephone book itself is not the set—it simply describes the set; it tells you in an organized way who is part of the group “New York residents” and who is not.
SET · A set is a pointer to a number of entities who share properties defined by the set (e.g., the set of the “Seven Seas” refer to the seven oceanic bodies of water of Earth, i.e., the four oceans and the three large Mediterranean seas). Put another way, sets are a way of organizing or grouping entities; they make life easier.
Sets are only enumerations of existing entities or other sets, not the entities themselves.
Set of all Sets
Why can sets not contain themselves?
One counter-argument that is often brought up against rationality is the notion of the existence of the “set of all sets.” Mathematics introduces its own axiomatic system which (unlike Objectivism) places artificial “axioms” first, rather than reality or the individual. At this point, Gödel’s theorem again becomes important: a system built upon axioms is necessarily either incomplete or inconsistent. Therefore, the fact that the construct “set of sets” can be defined, but due to its recursive definition (a recursive definition is a process, not an enumeration!) cannot exist, does not break with our considerations up to this point: mathematics can indeed (with recursion, if need be) describe any measurement (“complete”), but it is also possible to describe entities which cannot exist (“inconsistent”).
“Points” behave in a similar way in geometry (and the same holds for lines, cubes, etc.). Here again, we are dealing with representatives (pointers) because points do not exist; they are infinitely small, have no properties, and are representable in space only through imaginary relations and measurements.
Sets have to be countable and cannot contain themselves.
The simplest thought, like the concept of the number one, has an elaborate logical underpinning.
—Carl Sagan, Cosmos—The Lives of the Stars
Why is it that axiomatic systems in mathematics need not have a connection to reality?
In Chapter 1.5, “Ontology,” we discussed basic truths of reality. The three axioms mentioned—of Existence, Identity, and Consciousness—constitute an axiomatic system:
AXIOMATIC SYSTEM · An axiomatic system is the set of axioms that is the foundation of all knowledge within a field of study.
In mathematics, our discussion likewise deals with an axiomatic system. We now encounter the question of whether there could even be other axiomatic systems than that of Existence, Identity, and Consciousness. The axiomatic system of mathematics has arisen as a form of expression of measurements, which over time has become increasingly formalized. The more crucial difference between mathematical axioms and our axioms of philosophy is the fact that mathematics in most cases is practiced purely rationalistically. So the mathematical axioms used in it are not self-evident; they are only construed to facilitate particularly elegant depictions of numbers.
Axiomatic systems in mathematics need not have a connection to reality or be self-evident. They are purely rationalistic, self-contained systems.
The only limitation here is simply that the axioms of an axiomatic system cannot contradict one another (otherwise we could not create logical deductions of additional truths). There is an infinite number of such axiomatic systems. But as purely rationalistic constructs, they do not necessarily have to have a relation to reality (i.e., in a perception). A mathematical axiomatic system based on observations of reality would be a simple enumeration of situations and actions. For instance: “I have banana A; I put banana B beside banana A. Now I have banana A and banana B.” More complex statements or even general models of reality are not possible this way. Correspondingly, we have proceeded instead to attach mathematics to a system which is in itself free of contradictions.
What are possible shortcomings of a recursive description of natural numbers?
A simple example of a system not based on empirical experience would be the abbreviated recursive definition of the natural numbers:
- 1 is a natural number;
- If n is a natural number, then its successor is also; and
- 1 is never a successor.
By this definition, the statement “3 is a natural number” is true: 3 is the successor of 2, 2 is the successor of 1, and 1 is a natural number. These “axioms” thus do not contradict one another, but they are also simply plucked out of the air since they can be set up without any connection to reality. Here, the number 1 does not stand for an entity, but rather represents a measurement of the number of entities. On its own, a number is not an illustration of an entity or a set of entities. If we want to apply mathematical results in reverse, we must again allocate real entities to this abstraction. Consider the following example:
You would like to buy three bananas for your children. It makes no difference to them which bananas they are, as long as they have certain properties, that is, they can be allocated to the concept “banana.” Hence, the children would not accept an apple. It also makes no difference to them which child receives which particular banana.
Correspondingly, you would not say to the fruit merchant, “I want this banana for Tom, this one for Amelia, and this one for Peter”—you would simply say that you want to buy three bananas. From this statement, the fruit merchant does not know which banana is for which child, but he does not need to know—this shows the great power of abstract thinking: we save a lot of time by avoiding or disregarding specific cases. Conversely, the fruit merchant makes a measurement instead of asking you which banana you want. For this, he places the three bananas one after the other in a basket. Thus, he has concretized the mathematical result of the calculation and allocated three real entities.
Also, if we follow a number with successor after successor, we can arrive at arbitrarily large numbers. But this does not mean that something like “infinity” would have suddenly gained a foothold in reality as an entity. The axioms of the natural numbers describe a process whereby we can count any quantity of entities. But the fact that we can count up to a certain number does not mean that a corresponding quantity of entities actually exists in reality. Returning to our previous example, if you had asked for ten thousand bananas, the fruit merchant probably would not have been able to fulfill this request—such a number would not be available, even though the mathematics we defined allows us to ask for it. Thus, a systematic, logical mathematician (a rationalist) would get into conflict with reality time after time if he did not use sound common sense, i.e., if he did not translate the complete language of mathematics into a consistent one.
To illustrate, we cannot infer that if there are 10 oranges, there must also be an 11th or if it even makes sense to speak of 11 oranges. Think about other concepts such as days, spaces on a chess board, or truth values, whose quantities are all countable, while the natural numbers are countable but potentially infinitely so. A mixture of both forms of numbers can lead to errors if, for example, invalid values are entered in a form (e.g., the date “February 30,” or in chess, “move pawn to position space s11”—there is neither such a date on the calendar, nor such a position on the board). These errors indeed intuitively appear obvious to us, but in complex statements or philosophical systems this oversight can quickly escape us, which is why we have to be very precise in the wording of our definitions.
An erroneous mixture of complete and consistent systems also impedes us in our thinking. For example, if we regard the truth values True and False erroneously as a quantity (and assign the number 1 to True and 0 to False) instead of a set, we could ask ourselves whether there could also be a third or fourth truth value (like half-truth, quarter-truth, etc.) between or beyond 0 and 1. This question does not result from an observation of reality, but from a gap which is open because of our complete but inconsistent language. Just because we can enumerate entities of a set does not mean that it makes sense to ask what happens with larger quantities. Just because we have “1” life and just because we can count to 9, does not mean that it makes sense to ask about our lives “2” to “9.” Thus, such “mathematical” questions need to be checked first to ensure they make sense before investing any time solving them.
The recursive description of natural numbers indeed supplies a complete picture of reality, but, unfortunately, it is a potentially erroneous picture.
Why is mathematics not simply a science of entities?
As we established in Chapter 2.1.2, “Completeness and Consistency,” according to Gödel’s Incompleteness Theorem, axiomatic systems are necessarily either incomplete or inconsistent. As shown above, the recursive definition of the natural numbers indeed supplies a complete picture of reality (“complete”), but, unfortunately, a potentially erroneous picture, as there are (theoretical) representations of numbers which have no counterparts in reality (“inconsistent”). So, we have our philosophical axiomatic system (that is based on self-evident truths, i.e., it is consistent) on the one side, and the mathematical axiomatic system (that is constructed in a way as to be complete) on the other side. For this reason, mathematics ultimately is not the science of entities (like our consistent philosophical axiomatic system is), but rather principally the science of relations of entities, that is to say, measurements of their properties.
Mathematics is not the science of entities, but rather principally the science of relations of entities, that is to say, measurements of their properties.
An extension of the natural numbers are the so-called rational numbers, which we can generate recursively using the “Diagonal Argument” using the natural numbers, i.e. they are infinite but countable, exactly like the natural numbers. A rational number is thereby the ratio of two natural numbers, so from the outset it has nothing directly to do with a real, existing entity; rather, it is purely a mental concept which refers to a relation between entities.
The term “half of an apple” does not indicate an apple with a property of “half-ness” (as in the case of half of an apple hanging on a tree), but rather it should emphasize the temporal relation—the life history—of the apple: it was once whole and now has been halved. Alternatively, we can see this term as an instruction for the creation of the entity, the “recipe” for a half of an apple: “Take an apple and divide it.” Hence, if we use numbers and general mathematics, and we wish to forge a connection to reality, we should keep the idea of constructibility in mind.
We could rack our brain for a long time trying to figure out why a division by zero makes no sense. In light of our previous explanation, the cause becomes apparent: rational numbers do not always correctly depict reality, since their definition only refers to a relation between entities and not entities themselves. And since “nothing” is not an entity, division by zero is not constructible.
Did you know?
Mathematicians are divided on the issue whether constructibility—the so-called mathematical constructivism—is relevant for mathematical objects. This becomes important when discussing physics. Is it possible to follow insights about reality from mathematical relationships? → Read more in Philosophy for Heroes: Continuum
Just as the introduction of the irrational numbers […] is a convenient myth [which] simplifies the laws of arithmetic […] so physical objects are postulated entities which round out and simplify our account of the flux of existence […] The conceptional scheme of physical objects is [likewise] a convenient myth, simpler than the literal truth and yet containing that literal truth as a scattered part.
—Willard Van Orman Quine, On What There Is [Quine, 1961]
If the circle circumference is “irrational,” does this mean that there are no circles in reality? How do irrational numbers appear in nature?
The term “irrational” suggests that there is something in the universe located outside of our perception and our mind. It is true that we cannot represent the circumference of a circle using a natural number or a ratio of two natural numbers. Irrational numbers are infinite and uncountable since they, unlike the rational numbers, cannot be constructed from the natural numbers in a finite number of steps.
Irrational numbers do not refer to quantities or ratios and do not appear in nature as such. Instead, they refer to processes, or methods of generation (circles, golden ratio, leaf arrangements, proportions, etc.).
What aspect of nature is the source of its complexity?
We have already seen that rational numbers deal with an abstraction, namely relations between entities and their properties. With irrational numbers, the method of generation is simply one or more steps of abstraction deeper; so we are no longer dealing with abstractions, but rather with the results of measurements of properties of entities.
For the calculation of the circumference of a circle, we select a point with the distance of the circle radius from the center of the circle and then make infinitely many, infinitely small steps, changing our angle with each step by a correspondingly infinitely small value, so that we always travel around the center point at a distance equal to the radius. When we add the distances together, we obtain the value of the circumference.
But in practice, this method is not useful. While we would get an exact result, an infinite amount of time would be required to reach it. If we walk instead a finite amount of steps around the center of the circle, we get an approximation of the circumference. Figure 2.2 shows such an approximation with different polygons with an increasing number of corners within and outside the circle. A circle itself would constitute a polygon with “infinite corners.” This construction was first calculated and proven in the third century BC by Archimedes.
As this method of construction suggests, there can be, for example, no “circumference of a circle” amount of apples. A number can be constructed in completely different ways: just because something looks like a number does not mean that you can use it like an amount of entities. A number which represents a ratio is something completely different than a number which represents an amount. This is the reason it is important to use numbers in the right context and to know how they were constructed in the first place.
One conclusion from the fact that irrational numbers cannot be expressed as a relation (i.e., ratio) between natural numbers is that a multiplication of an irrational number by a chosen natural number always results in another irrational number. At first, this sounds like mathematical thought gimmickry, but upon closer inspection, we discover that nature makes extensive use of this quite abstract property.
The fundamental problem that a plant must overcome during its growth process is to get as much sunlight as possible on its leaves. If the leaves are arranged according to a regular (i.e., rational) pattern, such as “Leaf / quarter-turn / leaf / quarter-turn, …” the leaves will overshadow one another. The solution is to find an angle of rotation which can be continuously repeated so that no two leaves grow directly above one another (see Figure 2.3).
Nature’s solution is the so-called “golden ratio,” which can be calculated to arbitrarily high accuracy by the Fibonacci number sequence by the simplest means. In the case of plants, this number sequence is generated through cell division with a simple rule: “Each mature cell divides and new cells require a certain length of time to mature.” The process would occur as follows:
- From the new cell develops a mature cell. 
- From the mature cell develops a mature cell and a new cell. 
- From the mature cell develops a mature cell and a new cell, and from the new cell develops a mature cell. 
- From the two mature cells develops two mature cells and two new cells, and from the new cell develops a mature cell. 
- From the three mature cells develops three mature cells and three new cells, and from the two new cells develop two mature cells. 
As this sequence propagates, the Fibonacci sequence is generated: “1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,” etc., whereby the ratio of two successive numbers gives an increasingly accurate value of the golden ratio (1.618033…):
If we divide a complete rotation by this golden ratio, we obtain a series of angle values that minimize overlaps between leaves and thus maximize the amount of sunlight the plant can absorb.
If we draw squares using the Fibonacci numbers as the lengths of sides, we obtain the “golden spiral” (see Figure 2.4), which can be found in snails and flowers (see Figure 2.5).
We conclude that irrational numbers do not represent quantities of elements but instead are indicative of a very specific infinite process. The infinite process of describing a circle in terms of an n-sided polygon enables us to express the irrational number π; the infinite process of generating the Fibonacci number sequence is reflected in the golden ratio, and so on. The fact that we can speak about irrational numbers does not mean that reality is irrational—those are two completely different matters.
The complexity of reality stems from the fact that it is a product of infinitely repeating processes.
Did you know?
This golden ratio can be commonly found throughout nature (including human bodies). Likewise, our own perception is “calibrated” to find objects displaying this ratio to be particularly appealing. → Read more in Philosophy for Heroes: Act [Lode, 2018]
Mathematics and Empiricism
The real problem in speech is not precise language. The problem is clear language. The desire is to have the idea clearly communicated to the other person. It is only necessary to be precise when there is some doubt as to the meaning of a phrase, and then the precision should be put in the place where the doubt exists. It is really quite impossible to say anything with absolute precision, unless that thing is so abstracted from the real world as to not represent any real thing.
—Richard Feynman, New Textbooks for the New Mathematics [Feynman, 1965, p. 14]
What is the relationship between counting and philosophy?
If we gave an empiricist the (apparently simple) task of copying several lines from one sheet of paper to another by hand, he would be faced with a formidable problem: without the capability of counting, he has problems similar to those we would face if we had to draw a lawn. Would you start counting each blade of grass and then count again up to that number when making your own brush strokes? No, you would simply paint an area that just looks like a lawn—regardless of how the individual blades of grass actually are standing. And likewise, for an empiricist, a number of lines is but an image. He would not copy the lines from one paper to the other but draw a representation of “a number of lines.” [cf. Holden, 2004], [cf. Everett, 2012]
Counting is not a trivial capability or an innate one, but rather a deeply internalized expression of a system of philosophy learned early in life and embodied in language and culture.
Therefore, for the reasons discussed, there can be no mathematics in a purely empirically (i.e., in terms of empiricism) interpreted language, as quantities refer to recurrent concepts. [cf. Everett, 2009, p. 199] What we can learn from empiricists, though, is that we should not confuse numbers with entities. “One orange” is something principally different than “this orange here.” It is easy to mistake a measurement (“one”) of the number of entities of a concept (“orange”) with the actual entities (“this orange”). When an empiricist looks at a basket with six bananas, he cannot see 6 bananas. What he sees is the whole picture, this banana, that banana, that banana over there, etc. There is no plural for an empiricist. All he could do about the bananas is compare them to another experience he had, e.g., there are “‘more / less than usual’ banana” or “‘about as much as yesterday’ banana.” For him, each entity stands on its own and cannot be subsumed under a concept. Accordingly, for him there can be no word for “all;” that would certainly refer to concepts—all entities of this concept that exist somewhere. The difference between enumerations and measurements becomes important in the following section when we take a look at the number zero.
While there already was a depiction of zero in ancient Babylon about 4,000 years ago, they simply used it as a delimiter between individual numerals, like in our system the zero in, e.g., the number 101. At the end of the number, though, no zeros were added. If we applied this to our number system, it would be unclear whether the number 1 meant 1, 10, 100, or 1000. Here, you had to guess the scale from the context.
In the Roman numeral system, all numerals had to be added together, a system well suited for smaller numbers. With the calculation of distances in space by Indian astronomers in the 9th century, introducing the numeral 0 as well as the positional notation (e.g., 123 was no longer “1 + 2 + 3 = 6” but instead “1*100 + 2*10 + 3*1 = 123”) was required. This idea found its way from India to Europe by way of Arabic scholars and traders which is the reason why we, when speaking of our numerals, (erroneously) speak of “Arabic numerals” today (see Figure 2.6). Zero as its own digit was introduced in Europe only in the 12th century by Leonardo Bonacci (also known as Fibonacci).
Does “zero” have an equivalent in reality? If you have “zero” apples and “zero” lemons, is what you have (in terms of apples and lemons) the same?
Zero as an independent number was first actively used in Europe after the 17th century to apply to measurements of scales (temperature, sea level, etc.). With the emergence of rationalism and the desire to make a complete language out of mathematics, people integrated the number zero into existing number systems. From this, besides the problem of division by zero, there arose the question of what, e.g., “0 apples” means. What is the difference between “0 apples” and “0 pears”? What exactly happens if I buy “0 cows for 0 coins”?
When we consider the number zero today, we must keep this narrative in the back of our minds. Zero as a numeral is a blank space, and zero as a number is a measurement. To understand the concept of zero, you have to understand the difference between enumerations and measurements. If you point at a group of cows (“these five cows”), it is called an enumeration. If we notice that three cows are missing, it is a measurement (“minus 3 cows”). If we know that the cows named Berta, Elfriede, and Anja are missing, it is again an enumeration. Thus, the zero can never be an enumeration because we can neither point at it nor name it. That is the reason dividing the cake by zero guests does not work; a division by zero is forbidden in mathematics. The occurrence of a division by zero points to a deeper flaw in our understanding of the task in question: a confusion between an enumeration and a measurement. Thus, arguments that are substantiated by the number zero (or the “nothingness”) have to be examined extra carefully.
The digit “0” originally served merely as a blank space, while the number “0” has no identity and thus no equivalent in reality but can only represent the result of a measurement and is mostly used as a form of negation. You could count an infinite list of things that you own 0 items of, without ever making progress to describe what you own.
Mathematics and Reality
What are some of the limits of mathematics in terms of describing reality?
|potential entities||natural numbers|
|relations of entities||rational numbers|
|infinite processes||irrational numbers|
We can compare mathematics with the philosophy of entities (see Figure 2.7). Thus, mathematics can only be used as a language to communicate mathematical results. Mathematics itself has no direct relation to entities, mathematics only describes the relations and proportions of entities to each other, i.e., measurements of properties and enumerations. This missing connection to reality means that mathematics on its own cannot be used for philosophical statements. Also, even though mathematics can lead to new results in physics, this does not mean that reality is built upon mathematics. We cannot use a mathematical finding (e.g., the “existence” of irrational numbers) to then conclude something about reality. The fact that we can construct a perfect circle in an infinite number of steps and describe this process with an irrational number πdoes not mean that we live in an incomprehensible, “irrational” world.
Did you know?
To understand nature, it is not enough to perceive entities. The understanding of processes is a prerequisite to grasp concepts like science, consciousness, evolution, and quantum mechanics. → Read more in Philosophy for Heroes: Continuum [Lode, 2017]
Reality is without contradiction. Mathematics is a good tool to describe measurements of reality. But a “nice” mathematical model that reflects the results of measurements exactly still remains a model and is not necessarily a description of reality.