• Introduction
• Time and existence
• Existence and simultaneity
• A standard
• Units, arithmetic and identity: Experimenting on children
• Unequal "ticks"
• Clocks and causality
• Contextual standards
• Non-contextual standard?
• Can time speed up?
• Simultaneity at a distance
• Absolute time
• "Time dilation" and experiment
• Conclusion

These axiomatic concepts can be combined into formal axioms. Existence exists: there are things. Existence is identity: to be is to be something: things are what they are. (which is the law of identity) Consciousness is identification: to be conscious is to be conscious of existence. 

No proof can be offered for these, nor is any needed: it is self contradictory to deny them, for they must be accepted and used in every such denial. I.e., they are self-evident. (see Ayn Rand, Introduction to Objectivist Epistemology, Ch. 6) 

Axiomatic concepts and axioms are the root of all objectivity, of logic, and of honesty. Face it, there is no honest denial of the fact that things exist, or that they have some definite nature, or that one is conscious of them. 

We perceive a world of entities which come to be, move and change, and cease to be. These entities, changes and movements all exist, yet some exist, or last, or endure, more or less than others. Our identification of this more or less of existence—the "how long" of existence—is time. 

Every existent whatever can be measured by time. For example, time measures entities: that man has existed for 50 years. And so, time measures the quantities, qualities relations—and so on through every category—of those entities. In short, time is a measure of existence. 

This is based on induction. I invite you to review all of existence, trying to find a single existent which you cannot measure by time. You will fail to find one. 

It is often said (following Aristotle) that time is a measure of motion, but this misses the full generality of time. Aristotle himself was not misled by his formulation, for he went on to say that because time is a measure of motion, it is also a measure of rest. Since moving or not-moving, changing or not-changing, exhaust the possibilities for anything at all, Aristotle's view really boils down to the one stated here: time is a measure of existence. 

Commonsensically, by reference to the watch on your wrist you can measure the existence of anything within the range of your senses whether it moves or just sits there, e.g., that house across the street has been there for the last 5 minutes. Even a pre-school child knows that the answer to, "How long did that thing exist?," is always a number of time units. 

In fact, if you wished to specify the aspect of a motion which is most fundamentally measured by time, you would have to say that time measures how long that motion exists. 

This is compatible with Ayn Rand's remarks in Introduction to Objectivist Epistemology, Ch. 6. "The measurements omitted from axiomatic concepts are all the measurements of all the existents they subsume; what is retained, metaphysically, is only a fundamental fact; what is retained, epistemologically, is only one category of measurement, omitting its particulars: time— i.e., the fundamental fact is retained independent of any particular moment of awareness." 

Time is a measure of existence, but nevertheless we measure time by means of change, by means of things which come to be and pass away, and there is a reason for this. Something which existed unchangeably would provide no time markers for us to count, and measurement ultimately reduces to counting. 

We might compare, for example, the swing of a certain pendulum with a pulse beat, and perceive that the swing exists for as long as the beat. That is, we perceive that the swing came to be when the beat came to be, existed while the beat existed, and ceased to exist when the beat ceased to exist. We can abbreviate this by saying that the pendulum swing and the pulse beat are simultaneous. 

Time is a comparison of one thing's existence to another's. The most primitive form of this comparison is perceptual, and yields the concept of simultaneity. Simultaneity, like existence, is directly perceived. To perceive two things at once is to perceive them simultaneously. 

But perception of simultaneity is not yet measurement of time. We still have not said anything about how one measures time. I.e., we haven't yet derived time in the full sense. 

But we only arrive at the concept of time by selecting one particular (kind of) existent, and using it to measure all the rest. When such an existent is selected, it becomes the time standard. 

The standard is what gives time its uniqueness and universality: without a standard, we would have many "times," many different mutual relations between the durations of individual things. When we have a standard, we have one time which measures them all. 

Observe that no time standard exists in nature; a standard arises from a human choice. A time standard is an existent chosen by humans for the purpose of measuring time. Choice does not mean arbitrary choice: as we'll soon see, the choice of a standard is far from arbitrary. Just be aware that a standard is a contribution of human consciousness to the concept of time. 

We must establish an immutable standard. Why? 

The time standard must be immutable if you are to deal with time arithmetically; i.e., if you are to measure time. Why? 

If different units of time are not equal, then no arithmetic operations on them are valid. To so much as add or subtract times, you must know that the units are equal. If the units are not equal, the arithmetic results are meaningless: 5 seconds plus 2 hours equals 7 what? 

In fact, 1 plus 1 does not equal 2 unless both units are the same. If the units are not the same, arithmetic is useless. Let's explore this. 

It is proverbial that you cannot add apples to oranges, but the fact is you can if you know exactly what you're doing. Children know. When you ask them what is 1 apple plus 1 orange, they puzzle over it for a moment and then triumphantly announce: 2 fruits! And they're right! Philosophers should be so smart! 

The child's thinking goes something like this. I have to add two different things. But that doesn't make any sense; how could I express my answer? So I have to find something the same about them. What's the same about apples and oranges? Oh yeah, they're both fruits! The answer is 2 fruits. 

I performed this experiment on my own kids when they were very young, and they came through like champions! By giving them units which were progressively more and more dissimilar, I chased them up the ladder of all the genera they knew—all the way to "thing." 

The principle that you cannot add different units is fundamental. The identification of similar units is a pre-condition of arithmetic, without which arithmetic is invalid. 

In order to so much as count you must know that the things you are counting are similar. Without an element of similarity, you could come up with a "sum" of sorts but it wouldn't mean anything, it wouldn't be a sum of anything. In short, it would not be a sum after all, and the process you went through to arrive at it would not be counting after all. 

In order to count—i.e., in order to identify a sum—one must identify both a similarity in all the things counted, and how many of them there are. Otherwise one's alleged sum is not a number: it is a meaningless noise or symbol. 1 horse plus 1 cow equals 2 farm animals, but 1 horse plus 1 non-existent figment equals contradictory nonsense. 

We can look at this from a different angle. The number at which you arrive identifies only what is similar in the things you count. In counting you omit, or in a sense lose, all the knowledge about all the respects in which the things which you are counting differ. In counting apples and oranges together, you necessarily omit all knowledge about the ways in which apples differ from oranges, that is why you are obliged to express your result in some such form as 2 fruits. 

This may not seem to present a problem because we have a concept of last resort: existent. 1 horse plus 1 fleeting glimpse plus 3 sonatas plus 1 triangle equals 6 existents! But notice what has been lost in performing this ridiculous addition: all the knowledge about all the things which were added except for the bare fact that they exist; it's pretty thin gruel. 

The more similar the things you count, the more knowledge of them you can retain in your result. The less similar the things you count, the less knowledge of them you retain in your result. 

To sum up: To count is to count something. Every sum is a sum of something. A sum identifies only what is common to all the things counted. You must identify a unit which is common to the whole collection of things you wish to count in order to identify your sum. Counting, and therefore number and arithmetic in general, rest on the law of identity. 

What is lost in adding unequal units? All knowledge of quantity is lost. To identify quantity, we require equal units. Measurement is an extreme of counting, it requires units which are not only similar, but identical in the respect to be measured—i.e., they must be equal. 

It may be said that if you know the durations of two different ticks, you can add them anyway. But this begs the point at issue, namely how do we know durations. It pushes the problem back a stage; it assumes that you have some other unit which is equal in both the original units. You can add a day to an hour, but only by reducing them both to, say, seconds. Unless you can find an identical unit, you can neither measure nor add. Without equal units, you cannot identify quantity. 

Measurement is a very special kind of perception. To measure something is to count up equal units of it; the units must be defined before one can measure. Those units must be equal to one another if we are not to lose the quantitative knowledge we seek. Therefore, our standard of time must define equal units. 

Provided we define equal units, we lose no knowledge by selecting a single existent as a standard, and measuring everything else in terms of it. For then we can validly apply arithmetic and re-calculate the other ratios as needed. For example, if a certain pendulum swing is immutable, then it can be used to measure human growth, pulse beats, days and all the rest. The ratios of these things to one another can all be calculated from their ratios to the duration of the pendulum swing. 

We can define a standard and measure time without letting facts slip through our fingers if and only if our standard defines equal units of time. 

The basic difficulty is that you cannot directly compare successive time intervals. You cannot keep last week's second in a vault, and haul it out to compare with today's second. 

You compare successive time intervals by using a clock—and a principle. 

A clock is an entity which undergoes some repetitive process, which returns to the same state at the end of each period, which is in this sense immutable. 

The principle is that the same entity in the same state will act the same. The principle lets you infer that because the clock keeps returning to the same state, it acts the same, i.e., its ticks are equal. 

This principle is no ad hoc assumption; it is a form of the law of causality. In Ayn Rand's formulation, the law of causality states that: "All actions are caused by entities. The nature of an action is caused and determined by the nature of the entities that act; a thing cannot act in contradiction to its nature." (Atlas Shrugged, hard cover, pg. 1037) In a similar formulation, H. W. B. Joseph says, "If one thing the same in nature at different times, or two things the same in nature, are to act in situations the same in their nature, they must act on both occasions in the same way." (Introduction to Logic, Ch. XIX) 

These formulations display causality as a corollary of the law of identity. A thing acts as it does because it is the thing it is. Two things which are the same will act the same. 

It is important to grasp the full force of "same" in this law. When we say "two entities the same" we mean the same in all respects, not merely in all known respects. This does not imply that we claim omniscience. On the contrary, it stands as a reminder that we are not omniscient, that there may be unknown factors influencing any given process. 

Because things the same act the same, it follows that if two things which are the same in all known respects nevertheless act differently, that alone is enough to prove that in some unknown respect they are different. 

In this form, causality is immune to refutation. To refute it one would have to claim to have found two things identically the same which nevertheless acted differently. But only by claiming omniscience could one claim that they were indeed the same in all respects. 

Like the law of identity itself, causality enters our reasoning prior to any particular evidence and itself governs the interpretation of that evidence. It is as absolutely certain as the laws of logic, because it is a corollary of the fundamental law of logic, namely the law of identity. 

It is usually thought that prior principles such as this (prior, that is, to any particular experiment), interfere with discovery. But on the contrary, we see that causality directs our attention to what is currently unknown. When we observe differing actions of two apparently identical entities, causality tells us that there is a difference between them. The rational response is to inquire what that difference is, and to devise experiments to discover it. Far from hampering research, this "pre-conceived" principle inspires and directs it. 

Notice that the appeal to causality is implicit in the methods of time measurement actually employed; it is not some novelty foisted upon physics by philosophy. Physicists take immense pains to calibrate their clocks; they take for granted that identical clocks will tick out equal seconds; philosophy merely identifies this as an aspect of the law of causality. 

This may seem puzzling. If the standard is the base of time measurements, how is it possible to decide that the standard itself is wrong? 

Suppose the current time standard is a pendulum clock in a certain room. All measurements of time are ultimately counted up in terms of the ticks of this clock. But a very peculiar phenomenon is noted: every time the temperature of the room changes, all the rest of the universe speeds up or slows down in unison—out to the farthest galaxies. It is a universal non-local effect, appearing everywhere simultaneously. Clearly, either the clock or the entire rest of the universe is changing. Which one? We would soon conclude that there's something wrong with the clock, but what is the principle to which we would appeal to justify this conclusion? 

The answer, once again, is causality. There is nothing which changes all the rest of the universe simultaneously, but there is a change known to affect the clock, namely its temperature change. We happen to know that the period of a pendulum depends on its length, and that its length varies with temperature. Therefore the rate at which a pendulum clock runs varies with temperature. The "same" pendulum clock at a different temperature is a different clock. 

Even a physicist who did not know the effect of temperature on a pendulum clock could validly conclude that there is something connected with temperature which is messing up the clock—that there is some real effect altering the clock's action, that the clock is in some way different when it is at different temperatures. 

We can push this a step further. Even a physicist who was entirely ignorant of temperature could still conclude that something was altering the clock's action. In principle, this could even be a way to discover the fact of temperature. The alternative to deducing that an unknown real effect was altering the action of the clock would be to embrace a host of absurdities, such as the non-local effects mentioned earlier. 

A universal non-local effect is one which appears everywhere simultaneously. It would have to be mediated by a process which exists for no time at all. But whatever exists for no time at all does not exist. Non-locality is absurd in the full logical meaning of absurd. 

Furthermore, evidence would pile up that similar clocks made of different materials, or in different rooms, would produce different anomalies. Entirely different kinds of clocks, such as a quartz wristwatch, would have no anomalies or at least different ones. 

In light of all this, those determined to stick with the old standard would be faced with a culminating absurdity. They would have to claim that this particular clock, which they choose as a standard, governs every process in the universe, sometimes speeding them up and sometimes slowing them down. 

But if what they choose as a standard governs the universe, then their choice governs the universe. Thus, the alternative to acknowledging that an unknown real effect is changing the clock would be to suppose that their consciousness governs the course of the universe. The name for this particular absurdity is the primacy of consciousness, and it involves denials of identity and causality. 

Time rests on causality. It is only by reference to causality that we can select a standard, and measure time at all. It is only by reference to causality that we can decide whether an old standard may or must be abandoned in the light of new evidence. A given time standard is only valid within a certain context of our knowledge, and may have to be modified or replaced as our knowledge increases. 

In the first place, to claim to have found such a standard would be to claim omniscience. Such a claim would have to be prefaced by some such statement as: "I have reviewed all the causal influences in the universe, and have proved that none of them can ever, for all time to come, affect the rate of this clock." I think it unlikely that anyone will ever be able to validly make such a claim. No one now is entitled to do so. 

In the second place, consider the kind of clock which would have to underlie such a time standard. It would have to be truly eternal, absolutely unaffected by anything whatever, returning to identically the same state, again and again through all eternity. Before you start suggesting candidate eternal clocks, recall that the ancients thought they had one in the rotation of the heavens about the Earth. They had a lot to learn; so, perhaps, do we. 

A non-contextual time standard would not only be hard to establish, it would be useless for discovery. Before such a standard could be established, i.e., before it could be proved to be non-contextual, physics would have to be essentially complete, and provably so. This is implied by the omniscience claim mentioned above. Thus, such a standard could only be established after all the interesting facts had been discovered, not as an aid to such discovery. 

We do not have a non-contextual time standard, nor do we need one. All we have, and all we need, is a contextual physical standard, and the law of causality. 

To define time at all we must appeal to causality. Causality enables us to monitor the validity of our standard—and to amend it as necessary in light of new knowledge. 

We can expand a bit on his answer. Our time measurements are made in terms of a standard clock, which was chosen precisely for its immutability, so that all its ticks are equal. If evidence is found that something is making the ticks unequal, we must either replace the standard or apply corrections to it. In order to measure time, the rate of the standard clock must not change. 

The basic reply to the suggestion that time might change its rate is that time itself is not a rate, and does not have a rate. Only changes have rates, and time is not a change. Despite Newton's unfortunate metaphor comparing time to a river, time does not flow. Time is not a change, nor can time change. The concepts of "change," and "rate" are simply not applicable to time. If time changed it would not be time; time is not the kind of thing that can change. 

Then what the heck is it? 

Time is existence, measured by means of a clock, which was chosen for its immutability and validated by reference to causality. 

This formal definition rests on, and condenses, this theory of time. The emphasized terms point out the highlights. Existence is what we measure by time. Measurement is the human process by which we identify time. A clock is the physical instrument of time measurement. The clock's immutability is a logical pre-condition of time measurement. Its immutability is validated by the principle of causality. 

Observe that every item in this definition is essential to time. Drop any of them, and the concept evaporates. 

But observe also that all the complexity of the definition arises from spelling out the essentials of measurement. It is probably necessary to spell them out because there are so many theories which mangle them; but if we can take measurement as already understood, we get back to the simplicity we started with: namely, time is a measure of existence. 

So, can time change? To put it bluntly, time cannot change because we must not let it! Some may think that this implies assuming a godlike power over reality, but it simply reflects the fact that in measuring anything we must choose immutable units. It implies refusing to accept two seconds as equal if we know that the clock we are using was subject to different causal influences in the two cases. It implies refusing to falsify our knowledge. 

To suggest that time can change is analogous to suggesting that number can change. We accept without hesitation that if we put four rabbits on an island, the number of rabbits there can change, can become eight rabbits or no rabbits. We rightly regard it as absurd to suppose that four can become eight or zero! Similarly, there is no problem in accepting that any given process can change its rate, but to suggest that time can change is absurd. 

The meaning of time and number rest on immutability. If we allowed "four" to change, it would invalidate every calculation in which it appeared. If we allow "time" to change, it invalidates every equation in which it appears. 

Distant simultaneity is defined by the transport of clocks, and is validated by yet another appeal to causality: the same clock will act the same in different places. The validation of a mobile standard proceeds in essentially the same manner as validation of any other time standard. 

It is interesting to note that transport of a clock is exactly the method which was used by Captain Cook to carry Greenwich time all over the globe, and to map it. Until accurate mobile clocks were invented, the determination of longitude was very difficult. But by determining Greenwich time on the far side of the globe, navigators were able to determine that their astronomical observations were simultaneous to certain corresponding observations at Greenwich. It was then a straightforward exercise in geometry to determine their longitude. Accurate mobile clocks preceded the mapping of the Earth, by determining simultaneity at a distance. 

Time is universal, unique, and eternally uniform. We know it is because that's the way we made it. We had to make it that way in order to apply arithmetic; in order for our time units to possess identity. If we made it non-universal, non-unique or non-uniform, we would be fudging our units, and so would be unable to retain in our theories all the facts implicit in our perceptions. 

This validates absolute time, what is often called Newtonian time. 

Absolute time's anchor to reality is the law of causality, a corollary of the law of identity, the basic law of logic. As such, it is immune to experimental refutation. 

Certain kinds of clocks moving in aircraft or in Earth satellites run more slowly than the same kinds of clocks stationary on the surface of the Earth, and they slow down by an amount which depends on their fraction of light speed. Don't filter the facts through the theory of time dilation; just count the ticks! Those clocks really run slower. 

From causality, we conclude that the moving clocks are different in some respect from Earth-bound clocks, that something makes a difference in their actions. Since that difference depends on their motion, they are moving relative to something real. Let us call it vacuum, or, if you prefer, ether. 

Once we disperse the fog of time dilation, this establishes the existence of a real vacuum. 

Recall that it was the failure of the Michelson-Morley experiment to find convincing evidence for an ether which emboldened Einstein to decry ether as a metaphysical figment, and to develop his notion of time dilation. Now, evidence which clinches the case for real vacuum is taken as evidence for time dilation! 

This illustrates the principle that causality and absolute time are tools of discovery; theorists who have abandoned them are unable to interpret the evidence of those clocks. But their theories are wrong, time is absolute, and something is slowing those clocks. I wonder what. 

Second: Ayn Rand, who explicitly connected time to the axioms of existence, consciousness and identity. 

Third: Petr Beckmann, who woke me with the radical hypothesis that time doesn't dilate, that certain clocks do slow down. (Einstein Plus Two)