**History of Mathematics: **Its relation to CT 1

* “The book of nature is written in the language of mathematics”. –Galileo*

“*To be is to be the value of a bound variable*.” —*Willard Van Orman Quine *

“*However, I maintain that in any particular doctrine of nature only so much genuine science can be found as there is mathematics to be found in it”. — Immanuel Kant, Preface to “Metaphysical Beginning Principles of Natural Science” *

*Questions: Is absolute certainty attainable in mathematics? Is there a distinction between truth and certainty in mathematics? Should mathematics be defined as a language? What does it mean to say that mathematics is an axiomatic system? How is an axiomatic system of knowledge different from, or similar to, other systems of knowledge?*

Science as “the theory of the real”, the “seeing of the real”, is the *will* of this science to ground itself in the axiomatic knowledge of absolutely certain propositions; it is Descartes’ *cogito ergo sum*, “I think, therefore I am” . An *axiom* is a statement that is taken to be true, and serves as a premise or starting point for further reasoning and arguments. The word comes from the Greek *axíōma*: ‘that which is thought worthy or fit in itself’ or ‘that which commends itself as evident’. This “fittedness” and “self-evidentness” relates to the correspondence theory of truth, but it has its roots in the more primal Greek understanding of truth as *aletheia,* that which is “unconcealed” or “that which is revealed”. The axiomatic ground-plan or blueprint for all things allows the things to become accessible, to be able to be known, by establishing a relation between ourselves to them. But today, the relation of the knower to what is known is only of the kind of calculable thinking that conforms to this plan which is established beforehand and projected onto the things that are. Initially, this relation to things was called *logos *by the Greeks. The word initially meant “speech” or “communication”, but today it means “reason”, “logic” and is sometimes referred to as “theorems”.

If we use an analogy, we see the things as “data” or “variables” that are much like the pixels on a computer screen that require a “system”, a blueprint, a framework so that the pixels/data/variables can be defined and bound, and in this defining and binding the things are made accessible so that they can conform to something that can be known, some thing that we bring with us beforehand which will allow them to be “seen” i.e. the body of the bodily, the plant-like of a plant, the animal-like of the animal, the thingness of a thing, the utility of a tool, and so on. The blueprint or *mathematical projection* allows the “data” to become “objective”; the data are not objective until they are placed within the system or framework. If they cannot conform to the blueprint, the framework, the system, to this manner of knowing, then we consider them “subjective” and they somehow have less “reality”; they are not a “fact” because they are less “calculable”. One sees the effect of this framing in our language and the texting that is now a popular mode of discourse for us. Grave consequences are the result of the thinking that is bound by, and bound to, the “mathematical projection”.

The mathematical and numbers are obviously connected, but what is it that makes “numbers” primarily mathematical? The mathematics and its use of number and symbol that we study in Group 5 is a response to but does not ground our will to *axiomatic* knowledge i.e. the knowledge that comes from the axioms and the first principles that follow from those *axioms*. Modern mathematics, modern natural science and modern metaphysics all sprang from the same root that is the *mathematical projection *in the widest sense. It is within the “mathematical projection” that we receive our answers to the questions of “what is knowing?” and “what can be known?” i.e. to those chief concerns of our “Core Theme”.

The change from ancient and medieval science to modern science required not only a change in our conceptions of what things are but in the “mathematics” necessary to realize this change, our “grasping” and “holding”, our “binding” of what the things are, what we ourselves bring to the things. The change is one from “bodies” to “mass”, “places” to “position”, “motion” to “inertia”, “tendencies” to “force”. “Things” become aggregates of calculable mass located on the grid of space-time, at the necessity of forces which are partly discernible and with various predictable jumps across the grid that we recognize as outcomes, values or results. When new discoveries in any area of knowledge require a change in design (what is sometimes called a “paradigm shift”, but are not, truly, paradigm shifts), the grid itself remains metaphysically imposed on the things. This grid, this mathematical projection, is at the mysterious heart of what is understood as technology in these writings.

Modern Natural Science (physics, chemistry, biology) is dependent on mathematical physics. Modern Natural Science views the world through the lens of what is known as the “Reduction Thesis”: that there is a correspondence between science and the world, and that this correspondence can be demonstrated within the correspondence theory of truth using the principle of reason, the principle of non-contradiction, the principle of causality, and the principle of sufficient reason. *Science is the theory of the real. *The world, in ascending order of complexity, is composed of elementary particles (states of energy), higher, more complex, structures such as those observed by chemistry, yet more complex ones such as organisms that are observed in biology, and, lastly, human beings and their institutions (the Human Sciences). In a similar fashion, the sciences can be rank-ordered in a corresponding way with mathematical physics at one end and, at the other, the sciences concerned with the human: sociology, psychology, political science, among others which require more than simple mathematical results.

The status of mathematical physics (where algebraic calculation becomes authoritative for what is called knowledge) turns on its ability to give us an account of the *essential* character of the world (essence = its *whatness*), rather than merely describing some of its accidents (an “accident” is a “non-essential” category for what a thing is. You have brown eyes and I have blue eyes but these are “accidents” and have no impact on our both being, *essentially, *human beings). Can mathematical physics make such a claim i.e. does mathematical physics describe or give an account of what and how the world really is? its *essence*?

**Ancient and Modern Representation of Number:**

“Representation”, through the correspondence theory of truth, includes the conceptual tools which inform a world-view, or, to mix ancient and modern analogies, “representation” refers to the horizons, the limits defining this or that Cave, city,* nomos* (convention), civilization, or age. These definitions or horizons are the ‘paradigms’, ‘the stamp’ of what is considered to be knowledge in those Caves and determines what will be education in them. In the narrower sense, representation refers to the operations of the mind as it deals with concepts as well as its reflections on those operations, such as what we are trying to do here in TOK. We will examine the narrower sense here. We will note that the notion of a “concept” has been completely taken up in modern representation through imagination and reason, and these bring about the “knowing” and “making” that is the essence of technology. We shall try to do this with a reflection on the nature of number.

The Greek concept of number has a meaning which, when considered by First Philosophy (metaphysics), yields an ontology (the knowledge of ‘being-in-the-world’ and the beings in it) of one sort. The modern concept of number, on the other hand, while remaining initially faithful to this Greek meaning, yields an ontology or a way of being-in-the-world of a very different sort.

For the Greeks (and the tradition subsequent to them) number, the Greek *arithmos*, refers, always, to a “definite number of definite things”. Five or cinq or penta can refer to either five apples or five people or five pixels, but it must refer to a definite number of definite things. Alexander, one of the Aristotelian commentators, said: “Every number is of some thing”; the Pythagoreans said “The things are numbers”. As for counting per se, it refers to things or objects of a different sort, namely monads or units, that is, to objects whose sole feature is unity, being a “one”. For example, it would be as unthinkable for an ancient mathematician such as Diophantus to assume that an “irrational ratio” such as *pi*, which is not divisible by one, is a number as it is for us moderns to divide a number by zero. (The neologism, “irrational ratio”, only means a ratio which yields, in our terminology, an irrational number.)

Similar considerations hold for geometry. A triangle drawn in sand or on a whiteboard, which is an “image” of the object of the geometer’s representation, refers to an individual object, for example, to a triangle *per se*, if the representation concerns the features of triangles in general. For the Greeks, the objects of counting or of geometry are, if considered by the arithmetical or geometrical arts, in principle, incorporeal, without body. Hence a question arises as to their mode of existence.

Plato’s and Aristotle’s answers (whatever the differences between them, they are agreed on this) are that to account for what it means to say that there are pure monads or pure triangles must begin from the common ground which has been condescendingly called “naive realism” by the moderns. For Plato, pure monads point to the existence of the Ideas, mind-independent objects of cognition, universals; for Aristotle, monads are to be accounted for on the basis of his answer to the question “What exists?”, namely mind-independent particulars, like Socrates, and their predicates, that is, by reference to substances (*subjectum, *objects) and their accidents. An *accident*, in *philosophy*, is an attribute that may or may not belong to a subject, without affecting its essence. Aristotle made a distinction between the essential and *accidental* properties of a thing.

A few words on “intentionality” are needed here and to distinguish between first-order intentionality and second-order intentionality. We say that computers can be said to know things because their memories contain information; however, they do not know that they *know* these things in that we have no evidence that they can reflect on the state of their knowledge. Those computers which are able to reproduce haikus will not do so unless prompted, and so we can really question whether or not they have “knowledge” of what it is that we think they are capable of doing i.e. constructing haikus. They do not have “intelligence”, per se.

Much of human behaviour can be understood in a similar manner: we carry out actions without really knowing what the actions are or what the actions intend. Intentionality is the term that is used to refer to the state of having a state of mind (knowing, believing, thinking, wanting, intending, etc) and these states may only be found in animate things. In these writings these states are referred to as Being or *ontology*. Awareness of the thought of Being is the purpose of this TOK course and this may be called a “second-order” intention. So first-order intentionality refers to the mind directed towards those beings or things which are nearby, ready-to-hand. They are the concepts that we use to understand the non-mental or material things. Second-order intentions deal with abstract, mental constructs. Much discussion of this is to be found in Medieval philosophy in their attempts to understand Aristotle.

“First intention” is a designation for predications such as: ‘Socrates is a man’, ‘Socrates is an animal’, ‘Socrates is pale’. It not only serves as a designation for such statements or assertions about a thing, but it also characterizes their ontological reference or the ‘thing’ to which they refer i.e. to the being of what the thing is. Each of the predications listed above (man, animal, pale) has as an object of reference, a “first intention”; in Aristotelian terms a substance, in the Latin *subjectum* e.g., Socrates. It carries with it a “pointing towards”. (In this explanation, it is important to note language as “signs” in the word “de-sign-ation”. It is also important to note how our “reasoning” is based on the grammar/language of our sentences in English due to its roots in ancient Greek and Latin.) “First intentions” refer to our “first order” of questioning i.e. asking about the categories or characteristics of the things, their descriptions. We may say that the questioning about these characteristics is “first order” since they look at our assertions about the character of the the things and not about the thing’s “essence”. They are of the “first order” because they arise from our initial perceptions of the thing.

According to the Greeks number refers directly, without mediation, to individual objects, to things, whether apples or monads. It is, in the language of the Schools (the medieval Scholastics), a “first intention”. Number, thus, is a concept which refers to mind-independent objects. In order to understand the modern concept of number, it is useful to say a few words about the distinction between first and second intentions and show how these have come to be related to our understanding of “first order” and “second order” questioning.

With reference to representational thinking as understood by the ancients, not only is abstractness misapplied in this case of a ‘subject’ and its ‘predicates’, but the modern concept of number stands between us and an appreciation of why this is so. The Greek concept of number, *arithmos*, as stated in, say, penta, is a first intention i.e. it refers to mind-independent entities, whether it is apples or monads (things, units). The modern concept of number as “symbol generating abstraction” results from the identification, with respect to number, of the first and second intentions: both the mind-independent objects and the inquiring mind and its concepts are combined. It is what we have been calling the *mathematical projection *here. In order to make sense of the notion of a “symbol-generating abstraction”, we need to go to the modern concept of number.

Symbolic mathematics, as in post-Cartesian algebra, is not merely a more general or more abstract form of mathematical presentation. It involves a wholly new understanding of abstraction which becomes a wholly new understanding of what it means for the mind to have access to general concepts i.e., second intentions, as well as implying a wholly new understanding of the nature and the mode of existence of general concepts, and thus, a wholly new determination of what things are through a wholly new manner of questioning. This new ‘representation’ allows symbolic mathematics to become the most important achievement of modern natural science. Let us look at how this came about.

**Viete and Descartes and the New Understanding of the Workings of the Mind:**

In order to display where Viete departs from the ancient mode of representation, we need to focus on the use of letter signs and Viete’s introduction of letter signs into mathematics in the West. We think that a letter sign is a mere notational convenience (a symbol in the ordinary sense of the word in our day) whose function is to allow for a greater generality of reference to the things it refers to. But this use of symbols, as the character of “symbol generating abstraction”, *entails a wholly new mode of ontology or being-in-the-world and the representation of things of the world.*

Every number refers to a definite multitude of things, not only for ancient mathematicians but also for Viete. The letter sign, say, ‘*a*,’ refers to the general character of being a number; however, it does not refer to a thing or a multitude of things. Its reference is to a concept taken in a certain manner, that is, to the concept’s and the number’s* indeterminate* content, its variableness. In the language of the Scholastics, the letter sign designates a “second intention”; it refers to a concept, a product of the mind. But what is of critical importance: it does not refer to the concept of number *per se* but rather to its ‘what it is’, to “the general character of being a number”. The letter sign, ‘*a*‘, in other words, refers to a “conceptual content”, mere multiplicity for example which, as a matter of course, is identified with the concept. This matter-of-course, implicit, identification is the first step in the process of “symbol generating abstraction”. This step, which is entailed by Viete’s procedures and not merely by Viete’s reflections on his procedures, makes possible modern symbolic mathematics. In other words, at the outset, at the hands of its “onlie begetter” Viete, the modern concept of number suggests a radical contrast with ancient modes of representation.

For Plato and Aristotle* logos*, discursive speech/ language, is human beings’ shared access to the “content” of a concept, what was known as “dialectic”. It is through language, and as language, that mathematical objects are accessible to the Greeks. Not so for modern representation. The letter sign refers and gives us access to “the general character of being a number”, mere multiplicity (*arithmos*) (although it was left to Descartes to work out the implications of this mode of representation. More will be said on Descartes below.) In addition, the letter sign indirectly, through rules, operational usages, and syntactical distinctions of an algebraic sort, also refers to things, for example, five units. This leads directly to the decisive and culminating step of “symbol generating abstraction” as it emerges out of Viete’s procedures. It occurs when the letter sign is treated as independent; that is, when the letter sign, because of its indirect reference to things or units, is accorded the status of a “first intention” but, and this is critical, all the while remaining identified with the general character of a number, i.e. a “second intention”. Jacob Klein in *Greek Mathematical Thought and the Origin of Algebra* sums up this momentous achievement: a potential object of cognition, the content of the concept of number, is made into an actual object of cognition, the object of a “first intention”. From now on, number is both independent of human cognition (not a product of the imagination or mind) i.e. objective, and also without reference to the world or any other mind-independent entity, which, from the point of view of the tradition (if not common sense) is paradoxical.

What all of this means, according to Klein, is that “the one immense difficulty within ancient ontology, namely to determine the relation between the ‘being’ of the object itself and the ‘being’ of the object in thought is . . . accorded a ‘matter-of-course’ solution . . . whose significance . . . (is) . . . simply-by passed”. We can see now how the Quine statement beginning this writing (“To be is to be the value of a bound variable”) relates to this arrival of algebraic calculation. The mode of existence of the letter sign (in its operational context) is symbolic.

Let us try to grasp Klein’s suggestion about what symbolic abstraction means by contrasting it with the Platonic and Aristotelian accounts of mathematical objects. For Plato the correlate of all thought which claims to be knowledge is the mind-independent form, the “outward appearance” (*eidos*) and the idea (*idea*) or, in the case of number, the monad, the “unique”, singular one; none of these are the ontological correlates of the symbolic, modern grasp of mathematics. For Aristotle the object of the arithmetical art results from abstraction, but abstraction understood in a precisely defined manner. The abstraction of Aristotle is *diaeresis *where attention is paid to the predicates of things rather than the whole of a thing and the predicate is *subtracted *from the whole so that individual attention may be given to it. The subtracted thing has real existence outside of the mind.

The mode of existence of what the letter sign refers to in modern mathematics is not abstract in this Aristotelian sense, but is symbolic; it is more general. In the modern sense, both the symbol and what it refers to are not only unique, arising out of the new understanding of number implied by the algebraic art of Viete, they are, as well, logical correlates of one another, symmetrically and transitively implying each other i.e. such that, if a relation applies between successive members of a sequence, it must also apply between any two members taken in order. For instance, if A is larger than B, and B is larger than C, then A is larger than C.. That is, symbol in “symbol generating abstraction” is not a place marker which refers to some thing, as in the ordinary sense of symbol of our day such as a stop sign; rather it is the logical, conceptual, and thus quasi-ontological correlate of what it refers to, namely the “conceptual content” of the concept of number i.e. multiplicity. From this will follow (Newton) that all ‘things’ become ‘uniform’ masses located in ‘uniform’ spaces. The philosopher Kant will ground this viewing in his *Critique of Pure Reason.*

But at the same time, while bound to the ancient concept, the modern version is, paradoxically, less general. “Abstraction” in the non-Aristotelian sense, the label for symbolic modes of thought, can be grasped in at least two ways. First, it presents itself as a term of distinction as in the pair abstract/concrete. Whereas the concrete stands before us in its presence or can be presented through or by an image, the “abstract” cannot. Alternatively, “abstract” in the modern interpretation can also be illustrated by an ascending order of generality: Socrates, man, animal, species, genus. The scope of the denotation, or the extension, increases as abstractness increases, and, once again, the more general is also the less imaginable. But this is precisely what symbolic abstraction is not. The mathematical symbol ‘*a*‘ in context has no greater extension than the ancient number, say, *penta*. Rather, the symbol is a “way” or, in the modern interpretation of method which Descartes inaugurates, a step in a “method” of grasping the general through a particular (links to inductive reasoning and the scientific method may be made here as well as to the Greek understanding of *dianoia*). It is a way of imagining the unimaginable, namely the content of a “second intention”, which is at the same time through procedural rules, taken up as a “first intention”, i.e., something which represents a concrete ‘this one’. One consequence of this reinterpretation of the concept of *arithmos* is that the “ontological” science of the ancients is replaced by a symbolic procedure whose ontological presuppositions are left unclarified” (Klein, *Greek Mathematical Thought*, p. 184). What are the things which are represented here?

Descartes’ suggestion that the mind has such a power answers to the requirements of Viete’s supposition that the letter sign of algebraic notation can refer meaningfully to the “conceptual content” of number. The “new possibility of understanding” required is, if Descartes is correct, none other than a faculty of intellectual “intuition” (which we commonly call imagination). But this faculty of intellectual intuition is not understood in terms of the Kantian faculty of intellectual intuition. The Cartesian version, implied by Descartes’ account of the mind’s capacity to reflect on its knowing, unlike its Kantian counterpart, is not informed by an object outside of the mind. (Of course, since for Kant the human intellect cannot intuit objects outside the mind in the absence of sensation, there is no innate human faculty of “intellectual intuition”. It is, for Kant, a faculty that is impossible and illustrates a limitation on human knowing.)

Moreover, this power of intuition has “no relation at all to the world . . . and the things in the world” (Klein, p. 202). In other words, it is not to be characterized so much as either incorporeal or dealing with the incorporeal but, rather, as unrelated to both the corporeal and the incorporeal, and so perhaps is an intermediate between the “mind the core of traditional interpretations of Descartes. In the simplest terms, the objects of mathematical thought are given to the mind by its own activity, or, mathematics is metaphysically neutral; it says nothing about the being of a world outside of the mind’s own activities; it stresses subjectivity and subjectiveness.” The consequences of such thinking are immense and have been immense.

Nonetheless, this unrelatedness of mathematics and world does not mean that mathematical thought is like Aristotle’s Prime Mover merely dealing with itself alone. It requires, according to Descartes, the aid of the imagination. The mind must “make use of the imagination” by representing “indeterminate manyness” through symbolic means” (Klein, p. 201). A shift in ontology, the passage from the determinateness of *arithmos* and its reference to the world, even if it is to the world of the Forms of Plato, to a symbolic mode of reference becomes absorbed by what appears to be a mere notational convenience, its means of representation, i.e., letter signs, coordinate axes, superscripts, etc., thus preparing the way for an understanding of method as independent of metaphysics, or of the “onto-language” of the schools of our day. The conceptual shift from* methodos* (the ancient “way” particular to, appropriate to, and shaped in each case by its heterogeneous objects) to the modern concept of a “universal method” (universally applicable to homogeneous objects, uniform masses in uniform space) is thus laid down. Through this, the way is prepared for a science of politics (and all human sciences) whose methodology is “scientific” and to their reference within these sciences of human beings as objects and ‘masses’.

The interpretation of Viete’s symbolic art by Descartes as a process of abstraction by the intellect, and of the representation of that which is abstracted for and by the imagination is, then, “symbol generating abstraction” as a fully developed mode of representation (Klein, pp. 202, 208; cp. pp. 175, 192). Consider two results of this intellectual revolution.

1. In order to account for the mind’s ability to grasp concepts unrelated to the world, Descartes introduces a separate mode of knowing which knows the extendedness of extension or the mere multiplicity of number without reference to objects universal or particular outside of the mind. This not only allows, but logically implies, a metaphysically neutral understanding of mathematics. A mathematician in Moscow, Idaho, and one in Moscow, Russia, are dealing with the same objects although no reference to the world, generic or ontological, needs to be imputed.

2. “Symbol generating abstraction” yields an amazingly rich and varied “realm” (to use Leibniz’s sly terminology) of divisions and subdivisions of one and the same discipline, mathematics. For confirmation, one need only glance at the course offerings of a major university calendar under the heading “Mathematics”. Yet the source of this “realm” is at once unrelated to the world and deals with the “essence” of the world through mathematical physics in its essentialist mode. This is possible because the imagination is Janus-like. It is the medium for symbol generating and also a bridge to the world, since the world and the imagination share the same “nature” i.e., corporeality or, what comes to the same thing, the “real nature” of corporeality, extension.

Viete for one, as well as Fermat, simplified their achievements. They understood the “complex conceptual process” of symbol generating abstraction as merely a higher order of “generalization” thereby setting the stage for what has come to be habitual for modern consciousness, the passing over of the theoretical and exceptional, so that, in Klein’s phrase, it is simply “by-passed” or overlooked (Klein, p. 92). (All this is an inversion of Heidegger’s insistence that the passing over of the ‘proximal’ and ‘everyday’ must be overcome to appropriate Being in our day.) But this blindness to its own achievements, from which the modern science of nature suffers, is a condition of its success. Only if the symbol is understood in this way merely as a higher level of generality can its relation to the world be taken for granted and its dependence on intuition be “by-passed”. Only if symbol is understood as abstract in modern opinion’s meaning of the word would it have been possible to arrive at the bold new structure of modern mathematical physics on the foundations of the old.

It is important to grasp the conditions of the success of the modern concept of number. One of these is that modern mathematics is metaphysically neutral. This means, first of all, that modern mathematics does not entail, of itself, or presuppose of itself, metaphysical theses concerning what exists or what is the meaning of Being. For a contrast, one need only follow Klein’s patient exegesis of Diophantus’ Arithmetic; there, object, mode of presentation, scope of proof, and rigor of procedure are intermingled with metaphysics (Klein, pp. 126-49). Klein shows that “Aristotle’s theory … of mathematical concepts . . . was assimilated… by Diophantus and Pappus. Secondly, and more conclusively, the proofs and content of modern mathematical arguments need not be considered in conjunction with the metaphysical orientation of the mathematician presenting the argument, and so, whereas the pre-modern world could distinguish between Platonic and, say, Epicurean physics, no analogous distinction is viable in the modern world. There is yet a third way in which modern symbolic mathematics is metaphysically neutral and this in the strongest sense. It is neutral because it is all consistent with all metaphysical doctrines, nominalist or realist, relativist or objectivist. Whatever the metaphysics, to date, there is an interpretation of modern mathematics which leaves it unscarred. This is not the case for the ancient conception. For example, Euclid’s division of the theory of proportions into one for multitudes and another for magnitudes is rooted in the nature of things, in an “ontological commitment” to the difference between the two. Only after the metaphysical neutrality of the modern conception is taken for granted and bypassed, is it possible to do away with Euclid’s division as a matter of notational convenience.-

None of this holds true for mathematical physics in its authoritative mode, as arbiter of what there is (and what can, therefore, be claimed to be knowledge), in the version it must assume to serve as a ground for the acceptance of the victory of the Moderns over the Ancients at the level of First Principles (metaphysics). Mathematical physics does make in this mode metaphysical claims. It is not metaphysically neutral. Elementary particles are, for example, if mathematical physics is arbiter of what there is. But are they? One can see a corollary application of this thinking in the “objectlessness” of modern art.

Take, to begin with, the most influential version of ontology for those who accept the Reduction Thesis, that is, Willard Van Orman Quine’s famous dictum that “to be means to be the value of a bound variable.” Drawn as the dictum is in order to make metaphysics safe for physics, does it entail the existence of, say, elementary particles? All we know is that if we claim that particles are, that is, are in reality and not merely operationally defined then our claim will fit this semantic model. Conversely, sets, aggregates, mathematical infinities also qualify as “existents” in this semantic sense, but they cannot give us any knowledge of the world, since we need not impute to them any reference to a world outside the mind when we deal with them as pure objects of mathematics. In other words, as long as, in Cartesian terms, the identification of the real nature of body as extendedness with the objects of mathematical thought remains unproven and is merely, in effect, asserted, Sir Arthur Eddington’s hope that mathematical physics gives us an essentialist account of the world will remain just that, a hope.

All of the above means that Klein’s book is a key to understanding modernity’s most profound opinion about the nature of Being, of bringing to light the very character of these modern opinions in a manner which discloses not only their historical genesis but lays open to inspection why they are not only opinions but also conventions. Thus his book *Greek Mathematical Thought and the Origin of Algebra *is a key to renewing that most daunting of human tasks, liberating us from the confines of our Cave.