Category Archives: Science
In my previous posts, I discussed two critical questions about the rise of science in Europe in the 1400 to 1700 hundreds:
- Why there?
- Why then?
Let me begin with a older message by The Meeting House in the Greater Toronto Area that I watched on March 1, 2022 on YouTube (see Footnote), After describing the message, I will then show how it relates to the rise of science.
The message was part of a series entitled REASONS TO BELIEVE, and in this case was delivered by a guest speaker from Australia, Jerrod McKenna. It has nothing to do with science per se, but dealt with differences between the Greek view (the New Testament was written in Greek) and the Jewish view (most of the Old Testament was written in Hebrew) in understanding and interpreting the holy writings. Here are two figures adapted from the notes I took as I watched.
Figures 1 and 2
In Figure 1, the circle with the cross-hairs in the middle and the X at the center, represents the Greek view. The Greeks, valuing perfection, were always looking for the one perfect principle that unified. This is also the goal of science. Believing that these unifying principles exist is a major impetus for the search leading to their discovery. However, what happens if the “perfect principle” is not only imperfect, but wrong? The impetus that spawned the search for the perfect principle now becomes an impediment to changing it. When the data point arrives that destroys the beautiful law, one can always say, “Let’s put that data point in the filing cabinet until we know more. I’m sure with more data and more thought, it will eventually fit. After all, all my lectures and my reputations are built on the beautiful principle. If I claimed the principle is disproved, what would I teach?”
In Figure 2, the Jewish or Hebrew view is expressed, according to McKenna. The circle has two X’s on the periphery, representing two teachings or data points which are paradoxical, hard to reconcile, and from some perspectives, contradictory. Inside the circle is an area that could he termed “The Mystery of God.” In other words, one may encounter truths which are both true, but hard to reconcile (perhaps only at the moment). One can live with that because we are not God and cannot expect to understand everything. In other words, the Jewish view allows for uncertainty in the explanations. These are theological statements. How do the apply to science?
From Aristotle to Newton
A thorough description of Aristotle’s laws of motion has been presented:
The key one for our discussion is summarized in the figure above. Aristotle believed that natural state of terrestrial objects was no motion. In other words, to keep an object moving, one had to apply a force. This law is supported by observation a thousand times a day, by anyone who cares to look. You throw a stone, shoot an arrow, or kick a soccer ball, it moves for a while, slows down, and eventually stops.
The data that destroys the perfect theory usually comes before the new explanation comes. One has to live with knowing the theory is wrong and broken before one can describe what will replace it.
Aristotle’s laws of motion were seen to be incorrect, before the correcting explanations became apparent. Observing the four large moons of Jupiter clearly showed objects which did not come to rest. Galileo showed that some falling objects fall at the same rate independent of density. Quantitative estimates on how an object should behave were also not explained by Aristotle. But it took until the brilliance of Newton and his laws of motion, before an explanations emerged that overcame the problems.
Speaking as both a student and a tutor, I think one of the great failings in teaching science has to do with the false perception which leaves the student thinking that every question has been answered, and every science problem solved. It is much better to train the student to live with not knowing, or at least knowing that the principles we teach and talk about is likely fatally flawed, and we don’t yet know what the correct answer.
Summary and Final Comments
The philosophical climate in Europe in the 1400-1700 hundreds was precisely the climate necessary for the emergence of modern science:
- The Greek view of the perfect principle gave the impetus for finding a unifying explanation for data.
- When data came along that destroyed a well-established theory, the idea of The Mystery of God enabled scholars intellectually to realize the theory was wrong well before a better theory came along. Belief in The Mystery of God made it intellectually possible for them to say, “I really don’t know the correct explanation at this time. I know what we believed before was wrong. There are some things we may never know.”
- When a scholar is in a position where a much-beloved theory is discredited, yet no explanation has yet arisen to provide the new principle, one needs a bedrock of philosophic thought that allows this uncertainty to exist.
- The ability to say: “I don’t know” or “I no longer believe I know” is the scholar’s only sure defense against Confirmation Bias which makes it nigh impossible to dethrone a beloved, discredited explanation.
- The vivid imagination of pagan culture, which was carried over was an aid for rethinking explanations.
This discussion began with a book review of Peter Kreeft’s BACK TO VIRTUE. I hope this example was useful in understanding Kreeft’s and Meyer’s points in answering the question about the rise of science in Europe:
Footnote added: The messages in the WE BELIEVE series, at the time of writing, were no longer available on YouTube. If they become available again, I will add a hyperlink for the reader’s convenience.
One of my Facebook friends commented on this simple, apparently long-circulating, arithmetic problem and so it prompted many other of my Facebook acquaintances to also weigh in. The statistics of this FB post, as a whole, captured my attention. With 516K votes (“likes” I suppose) and 5.5M comments that is almost too impressive.
I don’t think the interest in this question would have been so high if everyone had arrived at the same answer. Since some did not, from the few answers I’ve seen, it seems many were prompted not only to correct the errors, but to also provide detailed explanations on the order of precedence rules in arithmetic.
Now, I’m naturally skeptical and suspicious (not always a good trait) and so I could not help suspecting that some of the wrong answers presented with conviction were nothing more than “click bait” and perhaps led to the phenomenal response to this simple post.
As a tutor in chemistry and physics this discussion provoked some interesting thoughts …
- In any math problem that involves a mathematical expression, the expression is a language that connects the person who set up and solved a math problem and someone else who uses the solution to find the correct answer to a similar problem. The conventions around the rules of precedence, that is to say: “multiplication and division must be done before addition and subtraction” are established so that the users of the equation understand how they are to perform the various operations correctly to get the right answer. If they are not followed, then using hyperbole, bridges will fall down, planes won’t leave the runway, and patients will received incorrect dosages. The rules involve shorthand (default rules that everyone is supposed to know) that make the expression as compact as possible.
- since multiplication is done first, as most respondents noted, the expression simplifies to:
- 50+50+0+2+2=? and so the answer is 104
- If the person who set up the expression wanted a different outcome, then brackets would be used to change the order of operations … (50+50-25)x0+2+2=? … this answer would be 4
- The conventions communicate from the person who set up the equation to the user. Like all conventions of this sort, they are only effective if we all agree to the same ones.
- The second interesting point has to do with calculators. Depending on the calculator, a person who has not bothered to learn the order of precedence conventions can easily get the wrong answer. Using a calculator is no guarantee of accuracy.
- For an older, simpler calculator that forces you to enter a number and an operation and another number to complete the operation, going from left to right will give the incorrect answer. By beginning at the left the user imposes an incorrect order of operation on the whole equation. You will in effect solve … (50+50-25)x0+2+2=?
- A more sophisticated calculator that lets you enter the whole expression will follow the rules of precedence
Bottom line: having a calculator does not necessarily keep you from making mistakes if you don’t learn the rules.
Since high school, one way or another, I have been associated with universities. First as a student (undergraduate and graduate), then as a Postdoctoral Fellow, as a research collaborator, and also as an Adjunct Professor. I have also participated in academic pursuits such as writing and refereeing papers. Organic Chemistry was my focus and through that discipline I met many fine people.
A writer of futuristic fiction is concerned about where things are headed
As a writer of futuristic fiction, I am driven by “What if …” questions. Since universities have played such an instrumental role in our culture in molding the sequential generations, naturally enough, some of the “What if” questions deal with trends or potential trends I have observed in higher education.
In my novel, The Halcyon Dislocation, the movement and isolation of a hypothetical University of Halcyon to a parallel world sets up an experimental literary sandbox. One can ask the question, what would the university elites do if they had the opportunity to channel the thinking of their students in any direction they chose? What would they choose? How would they get there?
What would university elites choose if they could mold student thinking in any direction they wanted?
One of the problems that plagues science, indeed culture and politics as well is the question:
If I can do something, how do I determine if I should do that very thing?
The “can” is usually determined by data, experimentation, and collective scholarship, but the “shoulds” remain elusive since they depend on the question of objective right and wrong which is inaccessible to data and experimentation. In the absence of an objective right or wrong, the answer often becomes: “Because I have the power and I want to, I will do it and no one can stop me.”
The danger then, for universities, is the tendency to becoming factories of conditioned students rather than nurturing educated students who have learned to thoughtfully consider opposing points of view in humility and respect.
Becoming factories producing conditioned students, rather than educational institutions that enable students to thoughtfully consider different viewpoints with respect, is one of the dangers universities face
The antidote to this tendency to become ever more efficient conditioners of students as our manipulative skills and technology increase, is to make sure opposing voices (including religious voices) are not only allowed to speak, but are heard and considered. Free speech is the best safeguard against conditioned speech.
A Recent Example That Hits Close to Home
I know of Organic Chemistry Professor Tomas Hudlicky by his fine reputation. He wrote, and had accepted a paper in Angewandte Chemie (along with the Journal of the American Chemical Society, Angewandte Chemie is arguably one of the two best journals in chemistry). However, after the Twitterati ignited a Twitter storm (Twitter Gewitter?) everything changed for Professor Hudlicky, According to an article in the National Post by Peter Shawn Taylor, the accepted paper was withdrawn by Angewandte Chemie, the two referees were taken off the referees list (I’m sure as volunteers they have better things to do with their time) and the editor was suspended.
I respectfully suggest you read the whole paper, as I did, or at least read up to page 4, along with Note 2 which seemed to cause all the offense and then think about discussing the points Professor Hudlicky is making.
The text of the paper if it’s still available … Hudlicky Paper
In my view, the proper way to proceed is to have everyone, first read the paper, then present their best arguments in respectful discussion. A view or position that is not permitted to be questioned, is likely indefensible. If the case for the other side were compelling, why not make it? Is that not the mission of universities to encourage students to properly discuss opposing points of view with respect and leave the final convictions that come out of the discussion to the students? Apparently not.
My Canadian public education, from elementary school, through high school and on through my university postgraduate studies, from the basis of inculcating a worldview, had a decidedly Materialistic bias. I was taught that all smart people were convinced by the overwhelming evidence of “science” that chance operating over billions of years produced “life, the universe, and everything.” They usually stopped short of explicitly stating that there was no room for God, but the extension of the teaching to this conclusion was easy and no barrier at all was set up to hinder this extension.
It was only in high school and university that I began to realize that a great many dubious philosophical presuppositions had been smuggled in with the historical assertions I had been fed. The many remarkable successes of what I now call “Good Science” were used to justify (if I looked at the data) “Dubious Science.” However, in the minds of most students, who had been taught to regard all science to be of equal value and veracity, the word “science” or “scientists believe” was used as a certificate of reliability.
Into this difficult and heavily contested discussion arena, Bruxy Cavey has provided his own input. Having listened to the first message on the first two chapters of Genesis, I think his goal is modest: he does NOT want to specifically argue for one interpretation or another, but rather to explore the language and context of the Hebrew text to provide a boundary to the range of interpretations that are consistent with the text.
Given that objective, I learned a few things.
One had to do with the Hebrew word Yom (day). It was interesting how it was used differently in the accounts of the seven days:
- Days 1-6 there was evening and morning cited after each creation event
- Day 7 , the Sabbath Day when God is resting from creation seems to go on without end. In Hebrews 4:1-11 we are urged to enter that rest.
- In Genesis 2, when the passage unpacks the creation of Man, the events such as naming the animals seem to require more than 24 hours.
Responding to Comments
I also wanted to interact with one of the interesting anonymous comments that appeared on Bruxy’s Blog. The comment is shown below in blue.
I was great at prayer and reading the bible when I was younger, but like so many, things changed when I went to university and studied science. Years later, I still love listening to science podcasts. I’m trying to reconcile what science says and what the bible says. I will never dismiss science because there is a lot to respect about the scientific method and the sweat, blood and tears that goes into understanding of the physical world around us, that is brought to us by relevant and worthy fellow human beings. While it can be said that science has just as much blood on its hands as religion, it has brought us the amazing technology I’m using to type this out, penicillin, the ability to “hear” remnants of the Big Bang and the understanding that a marble and a giant boulder will hit the ground at the same time when dropped from the same height (still blows my mind).
Sorry for belabouring the point on how much I enjoy science, but that’s not going away. And yet I want to make room for Jesus and his irreligious message. I love the focus on love and shifting my gaze from myself to others.
When I first heard that this series was coming, with special focus on Genesis, my initial reaction was “Uh-oh… this should be interesting.” While the stories seem to try and carry a message or lesson, I can’t take them literally…I just can’t. The only thing I can do to from dismissing them outright is telling myself that they’re essentially all symbolic, not to be taken literally; a way to try and explain something very complex in simple terms. Like trying to explain to a child why and how we do our taxes once a year…you can’t go into depth, so you sort of oversimplify and use symbols that they already understand; like, “we have to tell the mayor (to replace CRA or gov’t) how much money we made, this way they can decide if we give more or get some back,” etc. God is the alpha and omega: this, to me, means he’s like infinity, outside of the constraints of time and space. I can’t even understand what that would even mean, so how could I possibly understand how he actually started it all? Enter Genesis.
I guess I’m hoping for a Meeting House take on this and that I’m still allowed to show up
I will never dismiss science because there is a lot to respect about the scientific method and the sweat, blood and tears that goes into understanding of the physical world around us, that is brought to us by relevant and worthy fellow human beings.
We should all be truth-seekers since truth is connected to reality. While I understand the sentiment expressed by anonymous, science is not a uniform endeavor. Indeed, I think we ought to respect science by putting its best practices into operation as we evaluate the merit of a particular theory or claim. It all comes down to the data and the integrity of the people who collect and discuss it. Scientists, like other people, are confronted with political pressure, political correctness imperatives, natural biases, and peer pressure.
Even if we haven’t measured a data point, it still behooves us to be skeptical and ask the hard questions and see if the data adds up. Especially we ought to see:
- If the scientific community has tried hard to disprove the theory or hypothesis (it is easy to fall into confirmation bias and collect more and more data points in support of our favorite theory).
- If sufficient attention has been paid to data points that don’t support the theory. Or have they conveniently shoved the data into the “to be explained” file, never published, and promptly forgotten.
- If scientists are being pressure to adopt a certain view or theory. Look specifically for political pressure, political correctness imperatives, and peer pressure. Have scientists lost their jobs because of their hypotheses? Are there accusations of pseudo-science to keep you from looking carefully at the data and arguments? Have lectures been shut down? These considerations don’t over ride the power of the data but ought to cause us to dig deeper and find out what is being suppressed and pay particular attention to the voices that are being silenced.
it [science] has brought us the amazing technology I’m using to type this out, penicillin, the ability to “hear” remnants of the Big Bang and the understanding that a marble and a giant boulder will hit the ground at the same time when dropped from the same height (still blows my mind).
I generally agree. Notice, however, penicillin, and classical mechanics (i.e. gravitation and Newton’s Second Law) are qualitatively different from “the ability to ‘hear’ remnants of the Big Bang.”
The first category (isolating and characterizing penicillin or verifying classical mechanics) contain time-independent events and the critical experiments that can be reproduced in 2019, 2050, or 2200. The Big Bang is an historical event. A person with the proper resources can measure the background radiation, but they cannot perform the critical experiment (initiate a Big Bang and show it gives rise to the background radiation).
That doesn’t make the historical account incorrect, it just means the tools of scientific experimentation are not as well suited to these problems as they are to time-independent questions.
Anonymous wrote about reconciling what he has read in Genesis with the accounts that scientists propose:
When I first heard that this series was coming, with special focus on Genesis, my initial reaction was “Uh-oh… this should be interesting.” While the stories seem to try and carry a message or lesson, I can’t take them literally…I just can’t. The only thing I can do to from dismissing them outright is telling myself that they’re essentially all symbolic, not to be taken literally; a way to try and explain something very complex in simple terms.
That’s fair enough. My own reaction is somewhat different. I have significant personal experience that makes me trust what the Bible teaches. Still, as Bruxy stated, the Bible may be perfectly reliable, but that doesn’t guarantee my interpretation is correct. I line up all the historical theories of our origin side by side: evolution, intelligent design, and various creation theories and generate a plus/minus for each one. I think all theories have significant defects and so I am left with saying we don’t know the details.
Anonymous makes a very important point using his analogy of explaining the CRA to a child. Explanations are always constrained by the language and understanding of the audience. For me one of the great attributes of the God of the Bible: He reaches out to us. He uses the language and understanding of his audience to speak to us. I think we need to keep that in mind as we read Genesis.
I appreciate Anonymous’ comment and I appreciate my chance to interact with these ideas.
I once asked a friend of mine who reads a great deal of Science Fiction and Fantasy what he saw as the essential difference between the two genres. He thought for a moment and said that Science Fiction “could happen” while Fantasy “could not.”
I think I know what he meant. In Science Fiction, the writer is cognizant of the physical laws operative within the story. If an SF writer were to describe space travel, Newton’s Laws of motion and gravity would be obeyed. Even here one enters a grey area: some writers would insist on using the speed of light as a fixed limitation while others would imagine a way around it.
In my high school years, I grew up on this genre and my love of science, in large measure, grew out of that reading. Several friends had urged me to read The Hobbit and The Lord of the Rings, but I resisted for a long time. When I did read it, it was as if a new world had opened up for me. It recaptured for me what I had experienced as a child on first reading The Chronicles of Narnia. There was a sense of nobility, beauty, and “rightness” about those imagined worlds that I had missed in my Science Fiction reading, which instead, seemed sterile in comparison.
The longer I thought about it, it came to me that I was encountering an unspoken presupposition that was embedded in most SF literature, that of a materialistic universe where all that mattered was atoms and molecules; chemistry and physics. In addition, I found that the more modern SF also grew more cynical, growing increasingly hostile to the very things that I loved in Fantasy. As a consequence, I read very few modern SF stories (although I do try them once in a while) and spend much more time reading Fantasy.
So how has this impacted my writing? I think, in The Halcyon Cycle, I write Science Fiction that reads like Fantasy. I spend a good deal of time thinking about the physics and chemistry behind my imagined world (I think some of my readers would argue too much, in fact), but I also have many of the elements of a Fantasy story (swords, nobility, right and wrong which transcends worlds and physical laws for example).
Check out The Halcyon Cycle Books … http://bit.ly/2qzzi4P-Author
In The Halcyon Dislocation, I postulated the existence of time quantization as a means to setting up parallel, sibling worlds. Here is the description of this concept in an exerpt from the book:
Tired and hungry, Dave and Glenn returned to their room and turned on the TV to see if broadcasting had resumed. To their surprise Jennifer McCowan, the blonde talk show host of Halcyon Music, was on the air.
“Even without social media,” said McCowan in her gentle, lilting voice, “I know that everyone is asking ‘where are we?’ and ‘what’s happened to us?’ To answer those questions I’ve asked a friend of mine to the studio. Please welcome Vlad Sowetsky.”
Canned applause welcomed Vlad.
“So, Vlad,” said McCowan, “please tell our viewers what you do.”
Vlad, a tall, big boned youth in his mid-twenties, had a long, narrow face and close-set eyes, so that the overall impression vaguely reminded one of a horse. He had shoulder length hair and stubble on his face.
“To cut to the chase, I’m a graduate student with Professor Hoffstetter, and I was in the control room when the dislocation occurred.”
“So what actually happened during the accident yesterday?”
“Well,” said Vlad, “we were running the largest test on the force field to date. The plan was to—”
“Whoa,” said McCowan, “I think you are going much too fast. Tell the audience how the Hoffstetter force field works, but no jargon, please!”
Vlad screwed up his face as if he were being asked the impossible. “The force field appears as a bubble about the size of a soccer ball when we first generate it. The time inside the bubble is slightly behind our time. When we first make the bubble, the time delay—or offset—is very, very small so that the field is thin. That is to say, anything can cross it. We expand the bubble to the desired size and then thicken it. By ‘thicken’ I mean that we increase the time offset so the field begins to have an effect. First it stops large objects. If we increase the time offset even more, we could theoretically stop air molecules or light from crossing the force field boundary.”
“Field boundary,” said McCowan. “Now you’re lapsing into jargon again and losing me.”
“By field boundary I mean the edge of the force field bubble. Shooting a missile through this barrier is, as Hoffstetter would say, ‘like trying to shoot into last week.’” Vlad was beginning to get exasperated.
“Okay,” said McCowan, “please go on. Even if I don’t understand all of the physics, I’m sure there are many listeners who will.”
“Well, we had intended to expand the force field so that it enclosed the central building in the experimental area. However, while we were expanding the bubble, the first lightning strike overloaded the equipment and the expansion continued unabated.”
This was followed by a momentary pause and a baffled look on McCowan’s face. “How big did the bubble get?” she finally asked.
“I think it expanded to a sphere about four miles in diameter,” said Vlad.
“Then a second series of lightning strikes overloaded the offset controls, and the time offset increased enormously,” said Vlad. Beads of perspiration had appeared on his forehead.
McCowan uncrossed her legs and leaned forward. “Tell the audience what you think happened next,” she prompted.
Vlad took a deep breath. “I only have a half-baked theory. Do you know about quantization of energy?”
“Vaguely,” said McCowan, a blank look on her face.
“Let me see if I can make it as simple as possible. Macroscopically, that is, in the world of meter lengths and kilogram masses, energy seems to be continuous. It flows like a stream or a river. So if I ask how much energy it takes to lift this book,” he lifted a book from the table, “you can calculate the energy in joules to as many decimal places as you like. I can lift the book to any height and calculate the lift energy for each height. But when you go down in size, ten orders of magnitude to angstroms, the world changes. When lifting electrons away from the atomic nucleus, all the rules change, and one can only ‘lift’ the electron to discrete ‘heights,’ or energy levels. It’s like being able to lift this book in little jumps.” He demonstrated by rapidly lifting and stopping the book at various heights.
“Yeah, I know what you’re talking about. You’re bringing back unpleasant memories of first year chemistry. But what has that got to do with the Hoffstetter field generators and the accident?”
“Everything!” said Vlad. “I think time is also quantized.”
“You’ve lost me again. How can time be quantized?” asked McCowan. “And if it is, what difference does it make?”
“Well, think about it in relation to the quantization of energy that you learned about in first year chemistry. We think of time flowing past us like a stream moving at a constant rate. That may appear true in our macroscopic world, but what happens if, at very short time intervals, one reaches a minimum time (I call it a mintival for minimum time interval)? What if our existence at the time interval of a mintival consists of little jumps, like a jump second hand rather than a sweep second hand? Or putting it another way, what if instead of a flowing stream, time consisted of a series of pools,” and here he paused to let his words sink in, “and our existence is a discontinuous series of jumps from one pool to the next?”
“Your theory is fascinating, Vlad, but what has that got to do with the Hoffstetter field generators?”
“I just told you that the Hoffstetter field generators cause the matter inside the field to lag normal time by a very small amount, say ten to the minus thirty-second of a second—that’s a decimal point with thirty-one zeros after and then a one. Now let’s suppose…” Sowetsky turned and kneeled on the sofa and drew three contiguous rectangles on a white board behind his seat “…that these three rectangles represent three sequential mintivals in our world, or universe, if you like. Another world can coexist with ours, as long as the mintivals of that world are offset from those of our time.” He drew three more rectangles adjacent but offset to the first three, like bricks on the side of a building. “It would be like a single reel of film containing two movies, with the odd numbered frames representing our world and the even numbered frames representing another world. If two projectors played this interlaced film with one displaying the odd numbered frames and the other the even numbered frames, one film could give rise to two motion pictures. Similarly, although two solid objects cannot occupy the same space at the same time, they can occupy that space at different times, so to speak.”
“Keep going,” ventured McCowan doubtfully. “I hope our viewers are following you through all this.”
“Well, normally, when the Hoffstetter field generators shut down, they collapse back to the nearest quantized mintival. When the field generators overloaded, I believe we kicked over into the trailing mintival—hence the new world!”
“Well, I’ll be!” said McCowan, genuinely shocked. “Can we get back?”
“I don’t know,” said Sowetsky, frowning. “We only know how to make the Hoffstetter field lag time, not precede time. If we tried it again, we might jump into yet another world that lags this one!”
“You can’t be serious!” said McCowan.
“I’m deadly serious,” said Sowetsky evenly.
“We’re never going to get back, are we?” asked McCowan, her voice fading to a whisper as tears began to fill her eyes. She turned away from the camera for a moment. “I have one final question, Vlad,” she said, regaining her composure with obvious effort. “Did you tell Professor Hoffstetter about this possibility?”
“Of course! I told him not once but several times!” said Sowetsky. “That’s what burns me up so much.”
“What did he say when you told him?”
“At first he told me ‘science requires us to take risks,’ and finally he told me to stop raising the matter.”
Is this even possible? Normally in quantum mechanics, quantization comes about because of boundary conditions. Think of a guitar string. A loose guitar string doesn’t produce a pure tone. Only when it is stretched between two points (think boundary conditions) does one obtain a pure fundamental frequency along with the overtones. These frequencies represent quantization of the sound (the fundamental and overtones are related mathematically). It’s not easy to see why time should have boundary conditions and so quantization seems unlikely at first glance.
However, in 1899, the great physicist, Max Planck, proposed a natural unit of time based only on universal constants such as the gravitational constant, Planck’s constant, and the speed of light. Planck’s time (ca. 5.39 x 10^-44 seconds) is a small number indeed and is considered by many physicists as the shortest time interval possible. Similarly, the inverse quantity, 1/tp, is a frequency and may represent the maximum frequency possible. Perhaps there are boundary conditions for time and the idea of time quantization are not as far fetched as it seemed at first glance.
Relationship to Time Paradox
In any case, Planck Time, or the Mintival described by the character Vlad Sowetsky in The Halcyon Dislocation are very short time intervals indeed. They are much shorter than the time of one vibration of a hydrogen molecule or the shortest time observed experimentally (8.5 x 10^-19 seconds (2010)).
This provides a trivial solution to the time paradox. In the time paradox, one short circuits a chain of cause and effect events. That is to say, travelling back in time means the traveler invariably makes changes or initiates new causes that change the future. Or does he? There are usually two solutions. In one possible solution, each change initiates a new multiverse or parallel world strand.
A second solution (illustrated in C. S. Lewis’ The Great Divorce (he references getting the idea from a science fiction novel) centers on the idea that the traveler cannot affect the past at all. It’s like adamant. Not even a blade of grass could be bent by the visitor.
With very short time intervals, traveling backward in time does not generate a violation of the time paradox because over these time intervals nothing happens so nothing changes. So you see time travel should be possible as long as the trip backwards is very short!