What are Numbers?

Consider a common question: “how much is 7 + 5?” We can quibble about the phrasing: “how many are 7 + 5?” or “what is 7 + 5?” In any case, the answer is 12. But what is 12? We are asking about what a number is. What is a number? We use numbers all the time without considering what they are. We might well know what three apples are, or what three pencils are, but what is three? What is three, all by itself, when it’s not three apples or three pencils?

This question arises, in part, because of the representative nature of language. Words, whether spoken or written, are commonly understood to be symbols which refer to things. Nouns refer to persons, places, objects, or ideas. Verbs refer to actions or states of being. Adjectives and adverbs refer to properties and qualities; prepositions refer to relationships. What about numerals? When I see the numeral ‘3’ written on paper, it is a symbol, and I might well ask if it refers to something, and if so, what that something is.

If numerals refer to something, that something would be numbers. Numerals are physical symbols, consisting of ink on paper. What type of things would numbers be? Numbers would be non-physical things, i.e. metaphysical things, at least according to philosophers like Plato. Plato's view, adopted by a number of other significant philosophers over the centuries, is called dualism and realism. The word ‘dualism’ indicates that Plato is positing two levels of reality, a physical level and a metaphysical level. The word ‘realism’ indicates that Plato posits that mathematical statements are statements about an independently-existing state of affairs, that mathematical words refer to independently-existing things, and that mathematical propositions are true or false to the extent that the correspond, or do not correspond, to such an independently-existing state of affairs. Author Rui Vieira writes:

What exactly are the objects of mathematics, and how do they relate to our knowledge of them? Since Plato (427 BC–347 BC) such questions have been central to the philosophy of mathematics. Plato realized that mathematics seems to involve perfect circles, triangles, and so on. But as Plato also noticed, there are no perfect circles or triangles to be found in the world, only imperfect approximations. Imagine a polygon [shape] with an ever-increasing number of equal sides. As the number of sides approaches infinity, the polygon will become a circle. Thus a perfect circle may be conceived of as a regular polygon with an infinite number of infinitesimally small sides. So no matter how accurate our or our computer’s rendering of a circle may be, it will only be an imperfect approximation. Finite humans and their computers cannot create objects with infinite mathematical features, such as the infinite sides of our ideal circle. Plato concluded that since there are no perfect mathematical objects to be found in the world, the objects of mathematics – perfect circles, triangles, and indeed numbers themselves – must somehow exist as eternal abstract entities beyond space and time in some otherworldly Platonic heaven called the world of Forms (or Ideas). Plato’s particular type of mathematical realism (ie, of attributing objective reality to mathematical objects), has been one of the most prevalent views of mathematics among both philosophers and mathematicians ever since.

Opposed to Plato’s view, however, are several groups of philosophers, called variously ‘formalists’ and ‘conventionalists’ and including brilliant mathematicians like David Hilbert, Heinrich Eduard Heine, and Carl Johannes Thomae. These thinkers regarded numerals as non-referential. Phrased another way, they thought that numerals are numbers and numbers are numerals. A numeral, according to formalism, is a mark - ink on paper - which is related to other marks by means of a set of rules. Famously, the analogy to games arose. Consider a card game. The seven of spades, or the eight of clubs, does not refer to, or represent, anything. Its role in a card game is simply that if it is played, then certain other things must happen in the game.

According to the formalists, 7 + 5 = 12 because the rules of the game are such that when four symbols ‘7 + 5 =’ are written, then the fifth symbol must be ‘12’. By contrast, the mathematical Platonist considers that 7 + 5 = 12 because when the object called ‘7’ is added to the object called ‘5’, another object, called ‘12’, is formed.

The formalists - conventionalists are similar - thought that they had made a contribution to the progress of mathematics, because they had freed mathematicians from mysterious discussions of what type of ethereal thing a number might be. They felt that mathematicians should not have to be hindered by discussions of metaphysical objects.

The Platonists, on the other hand, saw that formalism contained certain weaknesses. If the formalist views were adopted as the foundations of mathematics, and those weaknesses emerged, the entire superstructure might collapse. A leading critic of formalism was Gottlob Frege, who wrote:

The question now forces itself upon us: Is calling these signs numbers enough to ensure that they have the properties of the numbers proper which we have previously been accustomed to regard as quantitative ratios?

To be fair to David Hilbert, he and some of the formalists did not make a blanket ontological assertion that there are no metaphysical objects to which numerals refer. Hilbert was not so much saying that numbers as Platonic objects do not exist as much as he was saying whether or not numbers exist as Platonic objects, we will go ahead and do mathematics based only on numerals as otherwise meaningless symbols manipulated according to the rules of mathematics. The only relevant meaning had by numerals, for Hilbert, is the meaning given to them by their place in the rules. Whether or not they referred to Platonic numbers, and whether or not those Platonic numbers really existed, was for Hilbert not relevant.

There is a detectable difference between formalists and conventionalists. Formalists treat numerals as meaningless symbols to be arranged and manipulated according to a set of rules. Conventionalists treat mathematical propositions, and perhaps also meta-mathematical propositions, as true by agreement: they are the stipulations of the mathematical community. But for the purposes of discussion the ontological status of numbers, formalists and conventionalists are very similar, and we may conveniently lump them into the same category.

In any case, Frege sees formalism and the formalist project as a failure. He alleges that it cannot hold to its own program. While the formalist lays the foundations of the mathematics promising to treat numerals and all other operators as meaningless patterns of ink on paper, and to simply do no more than rearrange those symbols according to a set of rules, the formalist will in fact later attribute meaning to those symbols. Frege asserts that the formalist, having kept meaning and reference out of mathematics at the foundations, will later sneak little bits of meaning and reference into mathematics in its more advanced stages of development. Frege writes:

This attempt at formal arithmetic must be considered a failure, since it cannot be pursued consistently. In the end numerical figures are used as signs after all.

Specifically, Frege noted the use of the operators ‘greater than’ and ‘less than’ and asked, in which way a numeral, as opposed to a number, could be greater or less than another numeral. True, the formalists can reply that numeral are assigned an arbitrary order, and that the operators simply reflect that order; but this line of thought is complicated by the fact that there is an infinity of numerals, in fact an uncountable infinity, and how would an order be assigned to them all? Likewise, the formalists wants to maintain that an expression like ‘3 ÷ 0’ is meaningless, but if the formalists have already declared all symbols meaningless at the foundation, in which way is ‘3 ÷ 0’ especially meaningless? Frege continues:

We saw that terms and expressions were borrowed unconsciously and without explanation from arithmetic that has a content (e.g. ‘larger than’ and ‘smaller than’) and that their role in the calculating game remained obscure, although it seemed to be highly important. Formal arithmetic proved unable to define the irrational, for it had only a finite number of numerical figures at its disposal.

Frege stands with Plato and rejects the formalist interpretation of mathematics. The matter is not resolved. The formalist camp will charge Frege with violating Ockham’s Razor. While Frege has succeeded in demonstrating that there are difficulties for formalism, he has not conclusively demonstrated the existence of Platonic numbers.

Language and Analyticity

Contemporary philosophy - or contemporary analytic philosophy, or late modern philosophy - has inherited from Kant, for better or for worse, a schema in which we divide propositions into those which are by their structure either analytic or synthetic, which are by their epistemology either a priori or a posteriori. Placing these dichotomies on a two-by-two grid, a system with four compartments results - a system which has seemed to some philosophers just a bit too conveniently tidy.

The question can be raised whether all propositions fit properly into one of these pigeon-holes, whether the dichotomies are exhaustive (is there a third option?), and whether the distinctions involved are absolutely binary opposites with no overlap or ambiguity between them.

One type of question which can be raised is about the nature of analytic propositions. The tautological nature of analytic propositions seems clear, especially when examples are taken from mathematics or abstract physics - the types of examples Kant gives in his Kritik der Reinen Vernunft. Yet more everyday examples from natural language can seem less clear. The question can be posed, whether Kant and others have selectively chosen their linguistic examples.

Philosophers, mathematicians, and logicians tend toward idealized versions of language when they look to investigate the structure of propositions. Gareth Evans writes:

Frege was the first to formulate a systematic theory of meaning for a fragment of natural language; systematic in that it sought to provide an explanation of how the significance of complex expressions, particularly sentences, depends upon the significance of their parts. Unsurprisingly, given Frege’s larger purposes in investigating the foundations of mathematics, the fragment which concerned him was free of many of the characteristic features of natural language; in particular, indexical expressions like ‘I’, ‘now’, ‘here’, etc. However, Frege did offer suggestions as to how his apparatus could be brought to bear upon such devices.

Language interested Frege, initially and primarily, to the extent that it bore upon his investigations into logic and mathematics. Frege’s ruminations upon language tended to restrict themselves to a small and overly tidy subset of linguistic phenomena.

Let us consider a more typical proposition, and the sentence which represents it: Lee Harvey Oswald is the man who shot JFK. While this proposition does not contain any of the words which are normally considered indexicals, its analysis will nonetheless be conducted with a view to the time in which such analysis in executed.

In the year 1900, neither of the men who are referents of terms in the proposition had been born - although there may have been, at that time, a man whose initials were JFK, and there might conceivably even have been a man named Lee Harvey Oswald. Yet in conducting an analysis, we know that whoever generated the proposition meant this JFK and not that JFK, and meant this Oswald, and not that Oswald. A whole host of problems arise with such natural language sentences, sentences which are quite messier than uncluttered sentences about “bodies having extension” or “7 + 5 = 12” and so forth. Questions arise about ostensiveness, about meaning, about proper nouns, about referents which do not exist, and about synonymy.

Although some philosophers have been hastily dismissive toward Frege’s symbol of assertion in his system, writing that assertion is a merely psychological phenomenon, one might yet acknowledge that Frege, however misguided his assertion operator might have been, was responded perceptively to the complexities which natural language offers.

Our proposition about JFK and Oswald will seem different if we analyze it in 1962, or in 1964. By a much later time - let’s analyze it in the year 2013 - it will be very much like what Kant calls an analytic sentence. Considered from the standpoint of the knowing subject, many knowing subjects know about Oswald only that he shot JFK, and nothing else. For such individuals, our proposition will seem almost like the quasi-mystical a = a of the post-Kantian German idealists.

Simply put: for many natural language sentences, the question of whether they are analytic or synthetic seems to be a temporal question and a psychological question. Its analysis may be different if such analysis is carried out before or after specific events. Its analysis may depend on the epistemological state of the analyzer - whether he knows certain facts.

The murkiness resulting from the analysis, or attempted analysis, of natural language sentences has caused some to doubt the framework of analytic / synthetic and a priori / a posteriori. Willard Van Orman Quine writes:

I am not satisfied that a clear general distinction has yet been drawn between analytic and synthetic. I am even more in the dark on the Kantian distinction between analytic and apriori. This much, nevertheless, I can say: If the statements of the usual higher mathematical logic are analytic, then so are such platonistic statements as ‘There are classes’, ‘There are numbers’.

In Quine’s book Word and Object, his linguistic example begins with a word which has a fully established place in language; the task of the linguist is to discover that word’s use and meaning. An alternative example might begin with the coining of a word: astronauts return from a distant planet, bringing a sample of an unknown substance. They hand it to a scientist in a laboratory, whose task is to analyze the substance. As the scientist takes notes on the sample, he has need to name it, for the mere purpose of writing his observations.

The scientist begins by naming the sample: he calls it the ‘ABCXYZ sample’ and uses this term consistently in his laboratory journals. We might ask the question: which sentences using the name ‘ABCXYZ’ are analytic, and which are synthetic?

At first, a small handful of simple sentences, mainly observations recording sense-data, will be analytic: “the ABCXYZ sample is solid at room temperature” or “the ABCXYZ sample has a density of so-and-so many grams per cubic centimeter.” As the scientist conducts his research, synthetic sentences will emerge as new and not-directly-perceptible information emerges: “the ABCXYZ substance is composed mainly of iron and silicon” or “the melting point of the ABCXYZ sample is so-and-so many degrees Fahrenheit.”

As data about ABCXYZ becomes more clear, it is printed in reference books, and eventually in textbooks. It is taught to undergraduate students and later even to high school students. Eventually, a generation of students arises, a generation which has never seen an ABCXYZ sample, or even a photograph of a sample, and doesn’t know the heroic tale of astronauts who brought it back from a distant planet. For this later generation, ABCXYZ has always been found in all reference books and textbooks. For this later generation, the proposition that ABCXYZ is composed mainly of iron and silicon may be analytic; indeed, it may be the only thing that this generation knows about ABCXYZ.

Quine discusses a similar example:

Suppose a scientist introduces a new term, for a certain substance or force. He introduces it by an act either of legislative definition or of legislative postulation. Progressing, he evolves hypotheses regarding further traits of the named substance or force. Suppose now that some such eventual hypothesis, well attested, identifies this substance or force with one named by a complex term built up of other portions of his scientific vocabulary. We all know that this new identity will figure in the ensuing developments quite on a par with the identity which first came of the act of legislative definition, if any, or on par with the law which first came of the act of legislative postulation. Revisions, in the course of further progress, can touch any of these affirmations equally.

If we accept Quine’s objections, we have at least two options: we might take a moderately Quinian approach, and say that any decision about a proposition’s analyticity is relative to time and circumstance. Along this route we might say that a proposition is at one point in time, for one analyzer, analytic, while at some other point in time, for some other analyzer, it is synthetic.

A radically Quinian - Quine himself apparently preferred the spelling ‘Quinian’ to ‘Quinean’ - position might abolish the distinction between analytic and synthetic altogether, declaring it illegitimate.

Newton and Descartes: Mathematics

Isaac Newton studied the works of most, if not all, of the significant mathematicians of his era and of earlier eras. When he began his studies at Cambridge in 1661, Rene Descartes had been dead for just eleven years; the Cartesian influence in mathematics, philosophy, and the natural sciences would have been detectable to the young Newton. Newton read the works of Descartes, studying them carefully; as is often the case in the history of philosophy, the former philosopher will have a twofold effect on the latter: Newton learned and adopted much from Descartes, but would grow to firmly reject other aspects of Cartesian philosophy and mathematics.

Newton’s first exposure to Descartes was probably not through a primary text - not through a book written by Descartes himself - but rather a secondary text, a book written by Frans van Schooten, a Dutchman who popularized Cartesian mathematics. One of Newton’s most famous phrases - about standing “on the shoulders of giants” - was written in reference to Descartes. Carl Boyer writes:

Early in his first year, however, he bought and studied a copy of Euclid, and shortly thereafter he read Oughtred’s Clavis, the Schooten Geometria a Renato Des Cartes, Kepler’s Optics, the works of Viete, and, perhaps most important of all, Wallis’ Arithmetica infinitorum. Moreover, to this training we must add the lectures that Barrow gave as Lucasian professor, which Newton attended, after 1663. He also became acquainted with the work of Galileo, Fermat, Huygens, and others. It is no wonder that Newton later wrote to Hooke, “If I have seen farther than Descartes, it is because I have stood on the shoulders of giants.”

Although Descartes earned lasting fame by securing a central role for algebra, not only in mathematics, but also in “natural philosophy” as the natural sciences and empirical observational sciences were then called, he had not yet progressed to the understanding that an algebraic equation and its curve plotted on a plane were equivalent, i.e., that having either of them amounted to having both, and that given either of them, the other was producible therefrom. Niccolo Guicciardini writes:

The seminal text in Newton’s mathematical formation is a highly abstract essay: Descartes’ Geometrie. He borrowed and annotated the second Latin edition (1659–1661) by Frans van Schooten. Here Descartes had proposed a novel method for the solution — he claimed in the opening sentence — of all the problems of geometry. It was on this text that Newton concentrated his attention. Descartes taught how geometrical problems could be expressed in terms of algebraic equations (this process was termed the resolution or analysis of the problem). He maintained that finding the equation and determining its roots, either by finite formulas or approximations, is not the solution of the problem. It was not a surprise for the contemporaries of Descartes and Newton to read that in order to reach the solution, one had to geometrically construct the required geometrical object. A geometrical problem called for a geometrical construction (a composition or synthesis), not an algebraic result. Traditionally, such constructions were carried out by means of intersecting curves. Thus, Descartes provided prescriptions to construct segments that geometrically represent the roots and are therefore the solution of the problem.

Modern readers will be aware that the Newtonian (and Leibnizian!) calculus enabled mathematicians to think beyond the limits of Cartesian algebra. Descartes was forced to use trial-and-error (‘heuristic’) methods, making at first only approximations, and then refining them.

By Newton’s day the heuristic method proposed by Descartes was labeled common analysis. It was contrasted with a more powerful new analysis, which tackled problems about tangents and curvature of curves and about the determination of areas and volumes that cannot be reached by the finitist means envisaged by Descartes. Common analysis proceeds by “finite” equations (algebraic equations, we would say) in which the symbols are combined by a finite number of elementary operations. The new analysis instead goes beyond these limitations because it makes use of the infinite and infinitesimal.

As soon as Newton began digesting Cartesian geometry, he began to ponder that algebraic equations might not only yield a curve, but rather also yield information about the curve. It is now commonplace for students in calculus to use the equation to determine ‘local minima’ and ‘local maxima’ using Newton’s methods.

Newton’s early notes on Descartes’ Geometrie reveal how quick he was in mastering algebra applied to geometry. In 1665 he began to think about how the equation could reveal properties of the curve associated to it via a coordinate system.

The need for information about these curves was both theoretical and practical. Drawing the curves with accuracy, and understanding how to derive information about those curves from their corresponding algebraic equations was simultaneously important both for concrete tasks like lens-grinding and for philosophical tasks in exploring the foundations of mathematics and physics. The reader will recall Newton’s interest in telescopes.

Descartes had devised several mechanisms for generating curves. In De Organica Conicarum Sectionum in Plano Descriptione Tractatus (1646), which Newton read in Exercitationum Mathematicarum (1657), van Schooten had presented several mechanisms for generating conic sections. This research field was connected with practical applications, for instance, lens grinding and sundial design, but it was also sanctioned by classical tradition and motivated the highly abstract needs underlined by Descartes. Newton was able to devise a mechanism for generating conics and to extend it to higher-order curves.

Cartesian methods were capable of managing only some types of curves, not all types of curves. Newton’s work in physics required that he have more powerful mathematics so that he could work with more types of curves.

In 1665, Newton deployed organic descriptions in order to determine tangents to mechanical lines, that is, plane curves such as the spiral, the cycloid, and the quadratrix that Descartes had banned from his Geometrie. The study of mechanical lines, curves that do not have an algebraic defining equation, was indeed a new, important research field. How to deal with them was unclear. Newton was able to determine the tangent to any curve generated by some tracing mechanism. He decomposed the motion of the tracing point P, which generates the curve, into two components and applied the parallelogram law to the instantaneous component velocities of P.

Descartes had simply been forced to omit the treatment of certain types of curves. The advent of Leibnizian (and Newtonian!) calculus has had a ubiquitous impact on the natural sciences. Carl Boyer writes:

Where Descartes’ geometry had once excluded all nonalgebraic curves, the calculus of Newton and Leibniz showed how essential is the role of these in their new analysis.

While the Cartesian approach and its ability to map algebraic equations on a plane still seemed new, Newton was already transcending it, and creating non-algebraic approaches. Niccolo Guicciardini writes:

For instance, the point of intersection of two moving curves will generate a new curve whose tangent Newton was able to determine. Such a method for determining tangents without calculation pleased Newton as much as did his new techniques for the organic description of conics. This was an approach to the study of curves — alternative to the Cartesian algebraic — that Barrow had promoted and that in the 1670s Newton began to couple with ideas in projective geometry. Already in 1665 the master in the common and new algebraic analyses was experimenting with non algebraic approaches to geometrical problems.

Leibniz and Newton broke through the boundaries of Cartesian algebra. Their calculus would fuel the industrial revolution, as more sophisticated steam engines and other devices would be designed with the aid of this newer, higher mathematics. But we might speculate that neither man had such pedestrian interests.

Using Leibnizian jargon, we can say that while differentiation of algebraic functions (accepted by Descartes) leads to algebraic functions, integration can lead to new transcendental functions. Newton referred to what are now called transcendental functions as quantities “which cannot be determined and expressed by any geometrical technique, such as the areas and lengths of curves.” Infinite power series — in some cases fractional power series — were the tool that young Newton deployed in order to deal with these mechanical (transcendental) curves.

Newton’s understanding of mathematics was colored by his interests in mystical, theological, and even occult understandings of science. He was convinced that ancient scientists had made marvelous discoveries, and hidden these discoveries in allegories and other cryptic writings. This famed “other side” of Newton enabled him to venture predictions on the exact date of the end of the world, which he envisioned in a fairly orthodox version of a triumphalist second advent - although in other matters, he was anything but orthodox, as his studies of Hebrew and Greek grammar led him to reject Cambridge’s Anglican understanding of the Trinity. (Newton’s prediction for the end of the world was in the year 2060.) Newton’s attempt to reconstruct what he held to be the lost knowledge of the ancients would lead him, he thought, to nearly miraculous or supernatural results.

Lucasian Lectures on Algebra stemmed from a project on which Newton had embarked since the fall of 1669, thanks to the enthusiasm of John Collins: the revision of Mercator’s Latin translation of Gerard Kinckhuysen’s Dutch textbook on algebra. Newton’s involvement in this enterprise was an occasion to rethink the status of common analysis. He began experimenting with what he understood as ancient analysis, a geometrical method of analysis or resolution that, in his opinion, the ancients had kept hidden. In his Lucasian Lectures on Algebra, which he deposited in the University Library of Cambridge in 1684 and from which William Whiston edited the Arithmetica Universalis (1707), Newton extended Cartesian common analysis and arrived at new results in this field. But even in this eminently Cartesian text one can find traces of his fascination with the method of discovery of the ancients. The ancients, rather than using algebraic tools, were supposed to have a geometrical analysis that Newton wished to restore. This was a program shared by many in the seventeenth century. He also made it clear that synthesis, or composition, of geometrical problems had to be carried on — contra Descartes — in terms wholly independent of algebraic considerations. The fascination with ancient analysis and synthesis, a better substitute, he strongly opined, for Cartesian common analysis (algebra) and synthesis (the techniques on the construction of equations prescribed by Descartes), prompted Newton to read the seventh book of Pappus’s Collectio (composed in the fourth Century A.D. and printed alongside a Latin translation in 1588). He became convinced that the lost books of Euclid’s Porisms, described incompletely in Pappus’s synopsis, were the heart of the concealed ancient, analytical but entirely geometrical method of discovery.

While Newton’s occult and mystical tendencies may have set him apart from some of his fellow philosophers and mathematicians, they did so by means of degree, not by means of kind. Robert Boyle spent time studying Hebrew grammar; Locke wrote commentaries on Pauline epistles, treating them straightforwardly as divine, inspired, and infallible. Like Newton, Descartes had looked to ancient authors for hints about geometry and algebra.

It seems that Newton did not know that Descartes expressed similar views in the “Responsio ad Secundas Obiectiones” in Meditationes de Prima Philosophia (1641).

While Newton saw mathematics as the structuring principle of the universe, he did not follow Descartes in formulating a rationalistic natural philosophy (in the sense of an a priori natural philosophy). Newton retained an empirical side. A central issue in Newton’s thought is the effort to harmonize the a priori with knowledge gained from experience. We can see this struggle in one of Newton’s most famous phrases, when he wrote that

propositions collected from observation of phenomena should be viewed as accurate or very nearly true until contradicted by other phenomena.

The phrase “very nearly true” is thought-provoking. In a common notion of truth as a binary opposition to falsehood, a proposition is either true or it is not. To be “very nearly true” is to be false. Yet Newton apparently thought that he was expressing something by writing this way. Perhaps this is the a posteriori at war with the a priori in Newton’s mind. Frederick Copleston writes:

That Newton attributed to mathematics an indispensable role in natural philosophy is indicated by the very title of his great work, the Mathematical Principles of Natural Philosophy. The great instrument in the demonstrations of natural philosophy is mathematics. And this may suggest that for Newton mathematical physics, proceeding in a purely deductive manner, gives us the key to reality, and that he stands closer to Galileo and Descartes than to English scientists such as Gilbert, Harvey and Boyle. This, however, would be a misconception. It is doubtless right to stress the importance which Newton attached to Mathematics; but one must also emphasize the empiricist aspect of his thought. Galileo and Descartes believed that the structure of the cosmos is mathematical in the sense that by the use of the mathematical method we can discover its secrets. But Newton was unwilling to make any such presupposition.

Newton morphed from being trans-Cartesian to being anti-Cartesian. He grew to see Cartesian notions as undermining his theological views. Scholars have debated whether Descartes, in his personal life, was a faithful and engaged Roman Catholic, or whether he was more nearly a deist who merely paid lip-service to Roman Catholicism. There is some plausible evidence on both sides. But in his philosophical writings, God is present in a necessary but impersonal way; whatever Descartes may have privately believed, his philosophy is nearer deism than orthodox Roman Catholicism. Newton, by contrast, posited a mystical or spiritualist type of natural philosophy, in which he sought interaction at the intersection between theological questions and his questions in mathematics, physics, and philosophy. Newton rejected Anglicanism; this rejection seems to have been sincerely motivated, because it created otherwise unnecessary obstacles to his work at Cambridge. Instead of Anglicanism, he developed his own belief system as he wrote commentaries on the books of the Old Testament and of the New Testament. Niccolo Guicciardini writes:

Newton intertwined this myth of the ancient geometers with his growing anti-Cartesianism. In the 1670s he elaborated a profoundly anti-Cartesian position, motivated also by theological reasons. He began looking to the ancient past in search for a philosophy that would have been closer to divine revelation. The moderns, he was convinced, were defending a corrupt philosophy, especially those who were under Descartes’ spell. Newton’s opposition to Cartesian mathematics was strengthened by his dislike for Cartesian philosophy. Descartes in the Geometrie had proposed algebra as a tool that could supersede the means at the disposal of Euclid and Apollonius. Newton worked on Pappus’s Collectio in order to prove that Descartes was wrong. He claimed that the geometrical analysis of the ancients was superior to the algebraic of the moderns in terms of elegance and simplicity. In this context, Newton developed many results in projective geometry and concerning the organic description of curves. His great success, achieved in a treatise entitled “Solutio Problematis Veterum de Loco Solido” (late 1670s) on the “restoration of the solid loci of the ancients,” was the solution by purely geometrical means of the Pappus four-lines locus. This result, much more than the new analysis of infinite series and fluxions, pleased Newton because it was in line with his philosophical agenda.

Newton was quite clearly opposed to any mechanistic view of the universe, deistic or atheistic, which saw the universe as running purely on physical principles. To which extent Descartes embraced such a view remains somewhat disputable; while Julien Offray de La Mettrie’s book Man a Machine probably goes further in that direction than Descartes himself would have gone, it is nonetheless plausible to see that Le Mettrie took his impetus from Descartes. In any case, Newton posited the existence of a Deity who interacts with the universe in tangible ways. Frederick Copleston writes:

He carried on the work which have been developed by men such as Galileo and Descartes, and by giving to the mechanical interpretation of the material cosmos a comprehensive scientific foundation he exercised a vast influence on succeeding generations. It is not necessary to accept the views of those who rejected Newton’s theological ideas and who regarded the world as a self-sustaining mechanism in order to recognize his importance.

The writings of the Dutchman Christiaan Huygens interested Newton: Huygens was applying the latest developments in mathematics in the construction of clocks and various astronomical measuring devices. The Dutchman’s mixture of interests was similar to the collection of topics on which Newton was working. Niccolo Guicciardini continues:

One should not forget another factor that determined Newton’s option for geometry in the 1670s: the encounter with Huygens’s Horologium Oscillatorium. In his masterpiece, printed in 1673, Huygens had employed proportion theory and ad absurdum limit arguments (method of exhaustion) and had spurned as far as possible the use of equations and infinitesimals (in his private papers he did employ symbolic infinitesimalist tools, but he avoided them in print). Huygens offered an example to Newton of how modern cutting-edge mathematization of natural philosophy could be presented in a form consonant with ancient exemplars. The Lucasian Professor immediately acknowledged the importance of Huygens’s work, and one might surmise that his methodological turn of the 1670s — which in part led him to cool his relationship with Collins and avoid print publication of his youthful algebraic researches — was related not only to a reaction against Cartesianism, but also to an attraction toward Huygens’s mathematical style.

While working in solidly a priori fields like pure mathematics, Newton was nonetheless drawn to the empirical observational sciences like astronomy. Thus he grew disenchanted with much about Cartesianism (if he had ever been enchanted). Descartes, by refining his a priori rules of thought, sought to arrive at an a priori natural philosophy. Carl Boyer writes:

In this he hoped, through systematic doubt, to reach clear and distinct ideas from which it would then be possible to deduce innumerably many valid conclusions. This approach to science led him to assume that everything was explainable in terms of matter (or extension) and motion. The entire universe, he postulated, was made up of matter in ceaseless motion in vortices, and all phenomena were to be explained mechanically in terms of forces exerted by contiguous matter. Cartesian science enjoyed a great popularity for almost a century, but it then necessarily gave way to the mathematical reasoning of Newton. Ironically, it was in large part the mathematics of Descartes that later made possible the defeat of Cartesian science.

The similarities and contrasts between Newton and Descartes did not arise in a vacuum. We have already seen the roles of Huygens and Hooke, Boyle and Galileo, Leibniz and others in the development of the era’s thought. A group of thinkers categorized under the heading of the “Cambridge Platonists” played a role in transmitting a Cartesian notion about the nature of space into Newton’s thought. Frederick Copleston writes:

For example, in his Enchiridion metaphysicum Henry More argued that the Cartesian geometrical interpretation of nature leads us to the idea of absolute space, indestructable, infinite and eternal. These attributes cannot, however, be the attributes of material things. Absolute space must be, therefore, an intelligible reality which is a kind of shadow or symbol of the divine presence and immensity. More was primarily concerned with arguing that the mathematical interpretation of nature, which separated the corporeal from the spiritual, ought logically to lead to the linking of the one to the other; in other words, he was concerned with developing an argumentum ad hominem against Descartes. But his argument appears to have exercised an influence on the Newtonian conception of space.

Thus one of the most distinctively Newtonian doctrines - Newton’s peculiar conceptualization of absolute space as independently existing - may have Cartesian roots.

Although it is probable that no philosopher is entirely consistent or coherent, given that all philosophers are human, it may be argued that Newton presents more internal tensions within his thought than some other philosophers. To which extent those tensions can be resolved, and Newton saved from internal contradiction, is a task for wise scholars. Yet it takes only a little charity to see how Newton viewed mathematics and physics, not as a coldly mechanistic belief system of deists and atheists, but as the handiwork of a present and personal deity who inhabits his universe and whose designs have a teleology to them. Niccolo Guicciardini concludes:

Newton encouraged his acolytes to pursue researches in ancient analysis and never missed the opportunity for praising those, such as Huygens, who resisted the prevailing taste for the symbolism of the moderns, the “bunglers in mathematics.” When the polemic with Leibniz exploded, he could deploy his classicizing and anti-Cartesian theses against the German. Thus, Newton’s last mathematical productions, publications, and (often anonymous) polemical pieces were driven by a philosophical agenda difficult to reconcile with his mathematical practice.

Perhaps the story - with its ambiguous interpretations - of Sweden’s Queen Christina is a fitting symbol for the careers of Descartes and Newton. While Descartes could seem rather detached from the Roman Catholicism which professed, Christina converted, upon the death of Descartes, to that faith; Descartes was the only Roman Catholic with whom she’d spent significant amounts of time. Her engagement with that faith seems to have been more passion than that of Descartes; she surrendered her throne for it. Likewise, Newton may have engaged more actively with some aspects of Cartesian thought, while rejecting other aspects of it, than Descartes himself did.

Looking for Life

One tenant of orthodox Darwinism is that life emerged, spontaneously, from lifeless matter. Understanding the implications of this proposition is essential to understanding the plausibility of evolutionary hypotheses in general.

Stanley Miller and Harold Urey conducted experiments in 1952 to test the credibility of the “warm little pond” hypothesis, so named because Darwin wrote:

But if (and oh what a big if) we could conceive in some warm little pond with all sorts of ammonia and phosphoric salts, light, heat, electricity etcetera present, that a protein compound was chemically formed, ready to undergo still more complex changes.

The Miller-Urey experiment, as it is commonly called, subjected mixtures of water, nitrogen and compounds like carbon dioxide, hydrogen sulfide, and sulfur dioxide to heating, cooling, and electrical sparks. Later versions of the experiment, conducted by other researchers as well as by Miller and Urey, varied the temperatures, times, and compounds. Eventually, if the Darwinian hypothesis is correct, at least one combination should yield life.

Since 1952, different variations on this experiment have been performed by different researchers. To date, no life has been created by this process. Among the different approaches are those which attempt to emulate conditions in the earth’s seas, and those which try to replicate the variables found in warm ponds found in the vicinity of volcanic activity. As Armen Mulkidjanian writes,

We attempted to reconstruct the “hatcheries” of the first cells by combining geochemical analysis with phylogenomic scrutiny of the inorganic ion requirements of universal components of modern cells.

Many of these attempts to create life have been presented as attempts to see whether amino acids, or other compounds popularly called “the building blocks of life,” could have spontaneously arisen in conditions presumed to have existed on earth. Even Miller and Urey, by the time they published their results, had recast their experiment as a hunt for certain compounds, rather than the hunt for life itself; the quest was based on

the idea that organic compounds that serve as the basis of life were formed when the earth had an atmosphere of methane, ammonia, water, and hydrogen instead of carbon dioxide, nitrogen, oxygen, and water

Having failed to grab headlines in newspapers - imagine “Scientist Creates Life in a Beaker!” - Miller and Urey had to content themselves with the comment that “this type of process would be a way of commercially producing amino acids.” They might be forgiven for their overly enthusiastic quest to create life in a test-tube: Miller was, after all, only 23 years old at the time. In a 1959 write-up, he framed it thus:

Since the demonstration by Pasteur that life does not arise spontaneously at the present time, the problem of the origin of life has been one of determining how the first forms of life arose, from which all the present species have evolved. This problem has received considerable attention in recent years, but there is disagreement on many points. We shall discuss the present status of the problem, mainly with respect to the early chemical history of, and the synthesis of organic compounds on, the primitive earth.

In the half-century, and more, since then, both the chemical content of the mixtures and the amount and type of energy input have been investigated, and variations of the experiment attempted. Researchers have used infrared, ultraviolet, and visible light; alpha particles, neutrons, and other forms of bombardments have been applied. The possible combinations of chemical elements, amounts and forms of energy, and the timings of different energy applications, mean that the number of possible experiments is infinite, or nearly so.

Although the names Miller and Urey were attached to this experiment, a researcher named MacNevin in Columbus had conducted similar experiments. A New York Times article, dated March 8, 1953, reports that

In Ohio State University laboratories Dr. Wollman M. MacNevin and his associates are re-creating the earth’s conditions two billion years ago - long before there was life on this planet. “One purpose of the study,” Dr. MacNevin explains, “is to answer this scientific question: Did extreme complexity of chemical compounds develop before life appeared, or was this a result of the life processes?”

Likewise, a researcher named Wilde published in the journal Science on July 10, 1953, another experiment which apparently predates Miller and Urey. Wilde writes, after sending an electrical current, in the form of an arc, through a mixture of water vapor and carbon dioxide, that

the action of radiation on these 2 gases is of special interest in relation to the basic photosynthetic process, and also carries implications with respect to the origin of living matter on earth.

Since Wilde, MacNevin, Miller, and Urey, papers have appeared at regular intervals, reporting similar results. The quest for “life in beaker” remains unsatisfied. What implications can we draw from what has been reported thus far?

Given that the number of experimental combinations to be explored is infinite, or nearly so, we will never arrive at the point at which we can empirically satisfy the desire to have tried every combination. To draw a conclusion, therefore, we will have to discern what constitutes a representative sample of the population. The population is every conceivable combination of elements and energy forms, including the variations in timing; other variables could be considered as well, e.g., pressure.

To have a representative sample of this population, we would need to divide the population into meaningful and significant categories, and draw samples from those categories. Recursively, each category would be organized into subcategories, etc., to ensure a broad range of samples. The question arises, would there be an infinite regression of subcategories within subcategories?

In this question, we encounter the larger question which inhabits all empirical inductive sciences. If a hypothesis is to be supported, or refuted, by empirical induction, and if there is an infinite or nearly-infinite population, at which point in the process of sampling do we consider that we have “enough” evidence to form a judgment?

Given that no mixture has been found which yields life upon absorbing certain types and amounts of energy, it might be argued that the Darwinian hypothesis has failed. But given that we have not yet tested all possible mixtures, it could be argued that the evolutionary schema has not been proved implausible. Neither argument is definitive until the problems of induction and sampling have been sufficiently clarified. This is the general problem of observational empirical sciences.

For the present, the question has been side-stepped, as scientists who are seeking to create life in a laboratory have learned to consistently present their work as merely the effort to determine whether or not certain complex organic compounds could have been produced in a lifeless environment. Despite more than sixty years of persistent effort, they have failed to demonstrate that a mixture of lifeless compounds can, upon the application of energy, generate life.

At the present stage, the only clear conclusion is that there is no evidence to persuade one to believe that life spontaneously arose from nonliving matter.

Not Quite Religion

What is religion? To answer that question, one will simultaneously answer the question what is not religion? By asking both questions, the possibility is raised that some things which are not religions might be called religion - might be mistakenly called religion, or mistaken for religion. Full-blown religions, like Judaism and Christianity, fall into that category, while other things, like magic, might more accurately be called part of a pre-religious phase.

Georg Wilhelm Friedrich Hegel, in his Lectures on the Philosophy of Religion, expresses a similar approach to religions and to those belief systems which are almost, but not quite, religions:

We shall discuss now the first stage of nature religion, the religion of magic, which we may deem unworthy of the name “religion.” In order to grasp this standpoint of religion we must forget all the representations and thought that we are perhaps so familiar with and that themselves belong to the most superficial habits of our culture. We must consider human beings all by themselves upon the earth, the tent of the heavens above them and nature round about them, and so, to begin with, without any reflective thought, altogether devoid of consciousness of anything universal.

Hegel points out that we need to undo our current cultural worldview, which is already informed by religion, in order to understand a pre-religious worldview. The more ancient civilizations embraced a worldview in which, in place of religion, one finds magic and myth. Magic, for these anthropological purposes, may be defined as an attempt to manipulate physical reality. Magic is the attempt to bring about certain states of affairs. One might use magic, e.g., to bring about fertile farming conditions or to bring about a military victory. Magic is thus closely related to, or identical with, what historians call “fertility religions” - but we note, with Hegel, that “fertility religions” are not religions, but merely so called.

Myth is the creation of narratives, narratives designed to explain. Myth can be true or false. In a magical, pre-religious society, myth is expanded and brought forth to do the tasks which religion will do at a later stage.

When a civilization leaves the stage of myth and magic, and progresses to religion, it abandons myth, because it acknowledges, with Kant, that sober reason admits that some questions are beyond it; human reason cannot answer all questions, and so mature religion is content to admit some mysteries. The immature stage of mythology does not want to admit that some questions are beyond the ability of human reason to answer, and so myths are fabricated, providing answers to all questions, leaving no question unanswered, and leaving no room for mystery.

Likewise, mature societies leave behind the phase of magic. Magic is the attempt to manipulate, and is not content to admit that humans cannot control every event in the natural universe. Progression from magic to religion is the progression from acting on the desire to place all variables under human control to acting on the recognition that humans must accept that there are natural limits to their powers.

Having shed magic and myth, the truly religious phase centers upon communication with the deity. The modern religious person, then, has a different approach to the realm of the spirit than did the person who belonged to a pre-religious civilization. Hegel writes:

It is difficult to get the sense of an alien religion from within. To put oneself in the place of a dog requires the sensibilities of a dog. We are cognizant of the nature of such living objects, but we cannot possibly known what it would mean to transpose ourselves into their place, so that we could sense their determinate limits; for that would mean filling the totality of one’s subjectivity wholly with these characteristics. They remain always object of our thought, not of our subjectivity, of our feeling; we can grasp such religions, but we cannot get the sense of them from within. We can grasp the Greek divinities, but we cannot get the inner sense of genuine adoration toward a divine image of that kind.

It is on this point that it is most difficult for the modern reader to enter into the psychology of early Mesopotamian cultures; into Mayas, Incas, and Aztecs; into Druids, Celts, Gauls, or Hittites; and into the earliest stages of Hindu, Greek, Roman, or Norse mythologies. It is telling that all of these engaged in the practice of human sacrifice. The drive for magic - the ability to manipulate the weather or to rig military victories - was so intense that human sacrifice seemed either necessary or reasonable. The modern reader may find it easy to mock those primitive cultures, but should remember that the immanent threat of death and the lack of accurate knowledge about the Infinite and the Transcendental are powerful forces; the modern reader may become more sympathetic when considering those forces. The modern reader might also imagine that certain aspects of modern civilization may seem equally barbaric when viewed by someone who is utterly outside that civilization.

Hegel’s Lectures on the Philosophy of Religion pose certain textual problems, being as they are, a patchwork quilt made of Hegel’s own notes from which he lectured, as well as notes taken by students during those lectures, along with fragments of text written out by Hegel - all edited together into prose by scholars after Hegel’s death. To compound the problem, the lectures, first given in 1821, were repeated over the years, in 1824, in 1827, and in 1831. Over those years, Hegel adjusted both the content and the form of his presentation, and editing all of this material into one stream of prose gives a simplified vision of Hegel’s thought on the matter.

Hegel began by noting that what one considers in this most primitive stage of culture, the culture of magic and myth, is not quite a religion, but rather something pre-religious. Yet he confusingly uses the word ‘religion’ in his discussion of it.

In the primal, immediate religion, here in this immediacy, humanity still knows no higher power than itself. There is, to be sure, a power over contingent life, over its purposes and interests, but this is still no essential power, as a universal in and for itself, but falls within the compass of humanity itself. The spiritual subsists in a singular, immediate mode.

The deities of early mythologies, and the power of magic, are then actually projections of the human mind. With Xenophanes, we note the suspiciously anthropomorphic features of these idols. Magic and myth are human ambition writ large. But even the formalization of these concepts into personified anthropomorphic deities is already a step beyond the most primitive level of myth and magic:

But the first nature religion is much more remote from the totality of our consciousness than this. Human beings in that situation still exist in a state of immediate desire, force, and action, behaving in accord with their immediate will. They do not yet pose any theoretical questions such as: “Where does this come from?” “Who made it?” and “Must it have a cause?” This inward divorce of objects into a contingent and an essential aspect, into a causative aspect and the aspect of something merely posited, or of an effect, does not yet occur for them. Similarly, even the will in them is not yet theoretical; there is not yet this rupture in them, nor any inhibition toward themselves. The theoretical element in willing is what we call the universal, right, duty - i.e., laws, firm specifications, limits for the subjective will. These are thoughts, universal forms that belong to the thought of freedom. They are distinct from subjective arbitrariness, desire, and inclination; all of the latter are restrained and controlled by the universal, or are conformed to this universal; the natural willing of desire is transformed into willing and acting in accord with such universal viewpoints.

Although his topic is the philosophy of religion, Hegel here thinks alongside various political philosophers who have imagined that humanity once existed in some ‘state of nature’ out of which society and state emerged. He is constructing a religious analogue to that political development. But both are dubious: politically and religiously, we must ask what evidence exists that humanity actually did live in that ‘state of nature’ and why we might not suppose that humanity simply lived in a less developed version of its current self. Those who posit a ‘state of nature’ are speaking of a difference in kind; we ask whether it might not have simply been a difference in degree.

But here human beings are still undivided with regard to willing; desire is the governing factor here. Similarly in their representations, in the imagination of these human beings, they carry on in this undivided state, this benighted condition, a stupor in the theoretical domain and a wildness of will. This is just spirit’s primitive and wild reliance upon itself. There is indeed a fear present here, a consciousness of negation, though not yet the fear of the Lord; it is instead the fear of contingency, of the forces of nature, which display themselves as mighty powers over against humanity. The fear of the Lord, which is the beginning of wisdom, is fear before a spiritually self-sufficient being opposed to arbitrariness. This fear first enters human experience when in one’s singularity one knows oneself to be powerless, when one’s singularity is inwardly shaken. The beginning of wisdom is when singular privateness and subjectivity sense itself as not being what is true, and, in the consciousness of its singularization and impotence, by way of negation, it passes over to knowledge, to universal being-in-and-for-self.

Hegel sees this pre-religious phase as the un-reflective and un-self-conscious activity of the will and of fear. He notes a transition from a fear of arbitrary contingent concrete fears to the the fear of the Lord. The themes of will and fear certainly continue from the pre-religious to the religious phase, but they appear in the latter in very different guise than in the former. One may also posit a post-religious phase, in which the primary concept is relationship with the deity, and not mechanism. In any case, when doing the philosophy of religion, it is a good starting point to sort out those things which actually are religion from those things which may seem like, or be called, religion without actually being so.

The Speed of Gravity

Posing a question about the speed of gravity yields an opportunity to examine questions about the philosophy of science, especially those which touch upon verification, experimental design, and sources of experimental error. Naively, we might say that Newton sees gravitation’s effects as instantaneous even at a distance, and that Einstein sees gravity as traveling at a finite, but not necessarily fixed, speed.

In terms of experiment and observation, astronomers have gathered data from phenomena like quasars and pulsars, and concluded that gravity is not instantaneous. Yet its speed is calculated by many of these astronomers as being significantly faster than the speed of light, a conclusion which, if accepted, would challenge many propositions taken as axiomatic by physicists.

While Newton took gravity to take effect instantaneously, he also considered any instantaneous effect at a distance as problematic. Many Newtonian thinkers after him solved the problem, or side-stepped it, by saying that the speed of gravity was so fast that it should simply be calculated as instantaneous.

We can conceptualize the question in a thought-experiment:

Let us imagine a universe devoid of matter except for three simple objects - e.g., three solid metal spheres, like ball bearings. These three are in motion, traveling in a plane. Although we are imagining a standard four-dimensional universe, i.e., the three Euclidean dimensions and time, the objects are moving only on a plane. They are in inertial states of motion as follows: two of them are on parallel paths, such that their routes would never intersect, save for the forces of gravity, and they will never leave the plane; the third is moving on a path such that it will collide with one of the other two. At any point in time prior to the collision, we can nicely predict their routes and project them into the future. The forces and their mathematics are well-known, and we have eliminated complications like friction or gravitational fields from other objects. The only quantities to be considered would be the masses of the objects, their speed and direction (i.e., velocities), and the forces of gravity between them. The two objects on parallel paths will presumably be drawn toward each other by gravity so that their paths would cease to be parallel. Let us stipulate that they will not collide until after the third object first collides with one of them, if ever. Up to this point, the situation is unremarkable, not controversial, and easily predictable by Newtonian means. At the point in time at which the third object collides with one of the first two, the two objects in the collision will change velocity. Let us stipulate that the collision does not destroy the objects; imagine billiard balls colliding on the table. Given that the collision and the change in velocity are either instantaneous or nearly so, we will regard this as a single point in time. The force of gravity upon the third object, the one not in the collision, will change also at this point in time. Given the change in the force of gravity upon that object, and the stipulated absence of moderating factors like friction, the third object’s path will change as the force of gravity upon it changes. Our question is then this: is the point in time at which the third object’s path changes different than the point in time at which the collision took place? If yes, then gravity is not instantaneous, and travels at a speed. If no, the gravity is instantaneous.

Philosophers are well aware that attempts to answer questions, whether successful or not, usually yield more questions and meta-questions. The present case is no exception. Is our thought experiment well-formed? Does it have any bearing on physical reality? It certainly fails to yield an answer to the question of whether gravity is instantaneous.

Any attempt to translate this thought experiment into observational natural science will be fraught with complications. Rather than working with a stipulated idealized universe, we would have to work with the real one. Attempting to observe and measure, e.g., tiny variations among asteroids, which travel through space with inertial motion and occasionally collide with one another, would be affected by numerous other gravitational fields in the area - fields emanating from other objects. Calculating these would be impossible or nearly so. The small changes in path and the precision needed to measure such small bits of time would exceed our technological grasp.

If there is any value attributable to this thought experiment, it might be this: that it presents the question in an isolated form. This is the procedure normally adopted by physics textbooks when presenting less controversial matter.

The question about the speed of gravity touches upon questions about the mechanism of gravity, and about the cause or source of gravity. Newton famously violated his own dictum that he would form no hypotheses. Commenting upon this benign self-contradiction, Professor Lawrence Sklar writes:

His hypotheses about gravity, for example, often have a very Cartesian flavor to them, as they postulate “ethers” that fill the universe with various fluid properties of pressure and resistance, and whose relation to matter (perhaps of lower pressure where matter is present, resulting in a “push” that moves matter toward matter) might, possibly, explain the law-like behavior of gravitational attraction. Such “mechanisms” might also remove from gravity the taint of action at a distance. It is worth noting here that the elements that later function to suggest the replacement of “action at a distance” theories by theories that propose an ontology of “fields” intermediate between the interacting objects, that is to say the time lapse in inter-particle actions and the violation in conservation of energy that results if one is not very careful in framing an “action at a distance” theory, play no role in the controversies embroiling Cartesians and Newtonians in Newton’s time.

Given that there are still unanswered questions and significant controversies about gravity, Newton’s suggestions, even if some of them are ultimately rejected, are still worth examination. Sklar continues:

Some of Newton’s hypotheses remain only curiosities in the history of science. Others, such as his particle theory of light, remain, if not really correct, important contributions to the development of later science. Still others, such as his hypothesis expressed in the “Queries” to the Opticks that there might be other forces along with that of gravity by which matter influences matter, and that these other forces might account for such things as the structure and behavior of materials, are prophetic insights into what became large components of the future growth of scientific understanding.

In post-Newtonian physics, questions about the speed of gravity take a different form. They might be posed as questions about the speed at which spacetime can change shape, i.e., curve or become curved. Or they might be asked about the speed of gravity waves, instead of the speed of gravity.

What is Life?

Aside from being a song by George Harrison, the question “what is life?” gives the philosopher occasion to engage in one of his favorite activities: working toward a precise definition for a word.

We use the noun ‘life’ and the verb ‘live’ (lives, lived, living) in ordinary circumstances quite often. Yet it is perhaps not easy to articulate exactly what life is.

One might observe that life is contingent: different forms of life are contingent upon different things. Various forms of physical life depend on various types of matter and energy: oxygen, hydrogen, carbon, heat, light.

We might inquire about non-physical forms of life. One hears the phrase ‘life of the mind’ - is that a metaphor, or a literal reference? Likewise with ‘social life’?

If we restrict ourselves to biological life, we might compare a dead organism to a living one. One of the differences is motion: dead things move when moved; living things move spontaneously. The motion of a non-living thing is ab alio; the motion of a living thing can be ab se.

Life is not measurable in the manner in which we measure matter or energy: a plant or animal does not lose any mass or weight when it dies. Viewing a single-celled organism through a microscope, we cannot see its life disappear when it dies: all the various parts of it remain, and remain in the spatial relation to one another.

What we can see is motion, and death is often inferred when motion stops - correctly inferred or incorrectly inferred. Motion is an indirect, or mediated, sign of life. Motion can fool us; we may think something is alive because it moves, and we might be wrong in so thinking. We may think something is dead because it does not move, and just as easily be right or wrong.

Life, then, might be something metaphysical. If it is not matter or energy or a combination thereof, and if it is not directly detectable by our five senses, it could be a metaphysical category. We note that it is not even detectable in principle by our five senses.

Whether or not one wishes to commit philosophically to a robust ontology, including a metaphysical entity or substance called ‘life’, one can see the reasons which would persuade a philosopher to make that commitment. We might speak of life “going” when a person dies. What is it that went? Whither did it go?

Although some texts include the phrase “self-sustaining” in attempted definitions of ‘life’, this could be true only in a rather limited sense. Given that life is contingent, it is sustained by the ongoing presence of certain types of energy and matter in its environment.

Although living beings may be capable of spontaneous motion, life itself is not spontaneous. Life does not start itself, but rather is started. It does not end itself, but rather is ended. Even in a case of suicide, life is ended by a means, mediately, not by itself directly.

Carl Christian Erhard Schmid wrote an extensive commentary on Kant’s philosophy, and published it in 1798. Summarizing Kant’s views, he gave a four-point Kantian definition of ‘life’ as follows:

1. In general: a substance’s power to determine out of an inner principle to act
2. Particularly: a finite substance’s power to determine itself toward change
3. A material substance’s power to determine itself toward motion and rest as changes of its condition. Because thought and desire are the only inner activities and principles of change which are known to us, but these are not objects of external senses, nor predicates of material substances, life is not ascribed to the latter
4. A being’s power to act according to the laws of the power of desire, i.e., by means of its representation to become the cause of the reality of the objects of these representations.

In German as in English, the verb ‘change’ has both a transitive and a non-transitive semantic; both are in play here. Schmid’s summary in the original reads:

1. Überhaupt: das Verögen einer Substanz, sich aus einem innern Princip zum Handeln zu bestimmen.
2. Insbesondre: das Verögen einer endlichen Substanz, sich zur Veränderung zu bestimmen.
3. Das Vermögen einer materiellen Substanz, sich zur Bewegung und Ruhe, als Veränderungen ihres Zustandes, selbst zu bestimmen. Da Denken und Begehren die einzigen uns bekannten innern Thätigkeiten und Principien einer Veränderung, diese aber keine Gegenstände der äussern Sinne, noch Prädicate materieller Substanzen sind, so kommt letztern eigentlich kein Leben zu.
4. Das Vermögen eines Wesens, nach Gesetzen des Begehrungsvermögens zu handeln d.h. durch seine Vorstellungen Ursache von der Wirklichkeit der Gegenstände dieser Vorstellungen zu werden.

Schmid’s original spelling has been preserved here. If Schmid’s understanding of Kant is correct - and we have reason to think that it is, because Schmid wrote during Kant’s lifetime and received no objection from Kant - then “life” is the power to cause objects to be real, according to Kant. That is quite a statement!