Embracing Leibniz, challenging Newton


In 1687, Newton introduced a new level of confusion about the finite-infinite.

An Easier, More-Simple Explanation

by Bruce Camber A first draft. First posted: Thursday, November 7, 2019 Last substantial update: December 9, 2019

Abstract: At one time big bang cosmology obscured a very simple model of the universe, especially our infinitesimal universe, that starts as a quiet expansion from the Planck scale. If base-2 exponential notation is applied from the first instant to the current time and size of the universe, there are just 202 doublings. Also, Isaac Newton’s absolute space and time obscured a finite-infinite relation whereby space-time and matter-energy are each derivative; continuity and symmetry, found throughout the universe, are the very definition of infinity and the necessary precursors of space-time and mass-charge. -BEC

Introduction. The people of this world have been confused about the finite-infinite.1 relation. Throughout most of our history, the experts around that discussion became known as religious thinkers. But, in 1687, on the 5th of July, a book, Philosophiae Naturalis Principia Mathematica,a authored by Isaac Newton, began teaching people how to become scientific thinkers. This book of about 1000 pages is filled with compelling philosophical discussions, mathematics-and-formulas, and many new concepts.

Among his new concepts was absolute space-and-time. If you look up into the clear night sky and say, “It goes on forever,” you could thank Newton for that conclusion. That’s absolute space and time and it’s now considered to be just commonsense by most people throughout the world.

Confident, profoundly intelligent, Newton possibly grasped more of the science of his day than anybody else alive at that time. A possible exception may have been a German polymath, Gottfried Leibniz. However, Newton, outlived Leibniz, and Newton had extraordinary position power:
• 1689–1701, Lucasian Professor of Mathematics, University of Cambridge
• 1689–1690, 1701-1702, member of Parliament for the University of Cambridge
• 1700–1727, Master of the Royal Mint
• 1703–1727, President of the Royal Society
• 1705, knighted by Queen Anne

In 1716 Leibniz died and necessarily conceded a two-year debate with Newton through his correspondence with Newton’s confidant, Samuel Clarke. Quickly published as a book,  The Leibniz-Clarke Correspondence.b, Leibniz argued for a relational understanding of space and time, but Newton’s absolutes prevailed.

Notwithstanding, the basic premise of this article is that the Leibniz observations about space and time are, even today, the easier, more-simple explanation of a key part of the finite-infinite relation. The universe (and space-and-time) has a beginning; and when you look up into the clear night sky, you see as far as the current expansion right now, today.

A primary assumption within this website. Most scholars recognize the depths of confusion within the sciences today. Although many excellent Leibnizian scholars have earned great respect, at no time has the Leibniz point of view about space and time been widely accepted. It would seem, given the many competing points of view within our time, that our leading scholars today would want to test the Leibnizian relational perspective.

We unwittingly backed into Leibniz in our high school geometry class. We went inside a tetrahedron.2 Like Zeno, we were stepping deeper and deeper inside when suddenly we hit a wall. Rather remarkably, Max Planck was there to teach us about his infinitesimals,3 particularly his units of space and time. It wasn’t easy, but once we began to understand the advent of the Planck units of a length and time, we began to grasp the realities of that wall. His imperfect calculations were the only “logical-possible” beginning of the physical universe. Space and time were relational and derivative, not absolute. It wasn’t a singularity, but a conference center of emerging relations between the finite and infinite.

We began to retrace our steps but immediately bumped into Leonhard Euler. He taught us about exponentiation. Doubling the duration and size with each step, we were at the current age and size of the universe, observing the current expansion, in just 202 doublings.4

We then compared. the epochs of the ΛCDM (Lambda Cold Dark Matter) model.5 with what could be described as a natural inflation.6 defined by our mathematically-integrated model of the universe. Our model readily absorbed the ΛCDM epochs. There was just a fraction-of-a-fraction of a nanosecond’s difference.

Our model may be an original.

Our model is easy, it works, and it is simple. With a bit more analysis, documents emerged to outline the fundamentality of relationalities:7  (1).Bottom-up,  (2) Map-the-Universe, (3).Transformations,  and  (4) Twelve Key Concepts.8 Our simple logic tells us, “The universe is exponential!” But what do we know? We’re just high school people. So we asked ourselves, “What could be more fundamental than space and time?” 

Current state of affairs. On November 4, 2019 distinguished astrophysicist, Joseph Silk, and his two colleagues, Eleonora Di Valentino and Alessandro Melchiorri (corresponding author), published Planck evidence for a closed Universe and a possible crisis for cosmology.c It is the result of their ongoing analysis of the cosmic microwave background radiation (CMBR),9 mapped by the European Space Agency’s (ESA) Planck satellite. And rather unwittingly, the Silk-diValentino-Melchiorri (SdVM) team looks at the data and concludes, “The Universe May Be A Closed System.” They concluded:

Planck cosmic microwave background spectra now preferring a positive curvature at more than the 99% confidence level. Here, we further investigate the evidence for a closed Universe from Planck, showing that positive curvature naturally explains the anomalous lensing amplitude, and demonstrating that it also removes a well-known tension in the Planck [ESA] dataset concerning the values of cosmological parameters derived at different angular scales. arXiv:1911.02087  (my emphasis)

Those keywords, a closed universe, were difficult to grasp. A flurry of articles quickly followed, however, nobody else would be analyzing that article in light of our emerging, rather simple model of the universe.10 Introduced in December 2011, our base-2 exponential model actually does mathematically encapsulate this entire universe within just 202 notations, but it is still expanding. It has no logical end. The finite-infinite relation is seemingly eternal.

Newton and Leibniz did not have the advantage of Leonhard Euler’s work in the 1740s when he first introduced exponentiation to our understanding of the functions of the universe. Nor did they have the advantage of Max Planck’s work in 1899 when he made his calculations that became known as the Planck base units.

My simple hope is that the SdVM observations and conclusions create a real debate that heats up, and eventually that debate becomes a definitive rip in the fabric of absolute space and time. Can we grasp and define the differences between (1) an interval of time, (2) the arrow of time, (3).the flow of time, and (4) the very nature of time?

Special groundwork.11 has been laid by many scholars from within different disciplines. From Loop Quantum Gravity, Carlo Rovelli says that time is an illusion. Then, from thinking about time travel, Berkeley physicist, Richard Muller, also calls time an illusion. From within string theory, Nima Arkani-Hamed says space and time are doomed. “Come up with something better!”

Butwe all experience time’s arrow and that experience is hardly an illusion. We all experience a second, a minute, an hour, a day, a year, and many years. Yesterday was yesterday, and we easily plan and plot and schedule for tomorrow. Within our human scale we experience the flow and direction of time. But on a macro-macro-macro scale, that is, in light of the entire universe, everything, everywhere, for all time, the flow of time may be relegated to an infinitesimal point that someday may be adequately explained. The best we can say right now is that all 202 notations are always active and always working with the others and the only time is NOW and everything-everywhere within space and time, is encoded within the universe and changes the universe.

Our Relational Model.12 First intuited in 1971-1972 and written out in 1994 and 2001 as the foundations of our foundations, this work initially began as a study of the concept of a moment of perfection to explain scientific anomalies. Now, those concepts have re-emerged, forty years later, but are now applied to the creation of time and space within 202 notations. We believe three concepts are precursors of space and time; these are continuity, symmetry, and harmony and are each found within the concepts of pi and spheres.

Space and time and space-time are always derivative but they are driven by the qualities of continuity, symmetry and harmony which cross the finite-infinite divide and are faces of both.


Your comments are encouraged.

For more, Bridging the Finite-Infinite-Divide, and On defining a Finite-Infinite Bridge

Endnotes & Footnotes

1 Finite-infinite.  Of course, to grasp the finite-infinite relation, we really need to know what is finite and what is infinite. It is a recurring discussion throughout the ages and on many pages on this website. It is a key concept. We ask anyone we can find who might have an answer (and who could readily understand our little chart of 202 notations and be able to tell us where we are going wrong). We started with many of the thought leaders of our time: Hawking, Guth, Penrose, Linde, and so many others. One of the early articles, Infinitude, further opens the discussion.

2 Tetrahedron. The simple tetrahedron is not sufficiently valued. We developed a special class just to teach the Platonic solids using clear plastic models. We thought it all amounted to a wonderful STEM tool and we began to share the information with other secondary schools. Yet, when we found no references to a base-2 chart of the universe, we stopped. We did not want to be spreading half-baked ideas that could taint our students in their university programs. Back in 2012, John Baez, a physicist  from the University of California-Riverside, was one of the first to tell us that this work was idiosyncratic. We quickly learned how right he was!  More

3 Infinitesimal units of space and time. Defined simply as things too small to be measured, the first 64 notations of the 202, indeed, are much too small to measure. Yet, their base-2 values from the Planck base units can be readily calculated. All are below the thresholds of measurement. To help interpret the data, there are lots of people to help us: Archimedes, Nicholas of Cusa, Kepler and even Leibniz.

We obviously have a lot to learn so we began a notation-by-notation chart of Planck Length (space) and Planck Time. And although it has a long way to go before it becomes robust, it still opens a construct for thinking about structure and the very nature of structure which here involved basic geometries and functions like π (pi) and the dimensionless constants involved within the definitions of the Planck base units. Heretofore, these simple concepts had not been keys to understanding the very start of the universe. Many asked, “Why now?

4 202 doublings. The chart of 202 doublings has not been publicly analyzed by any one of the hundreds of scholars from whom we have requested help. Generally, our best scholars are rather quick to help high school people to keep them on the straight and narrow. Yet, such advice never came. It seemed to be indirect confirmation that we just might be working with a new concept; we began to think, “Perhaps base-2 has never been applied to the Planck units.”  Being quite naive, we readily ventured into the unknown. We didn’t have old assumptions to cling to like Newton’s absolute space and time or big bang cosmology…

I have often asked myself, “Why hold on to things that have served us so poorly? The big bang theory is inherently nihilistic.” All of that became part of the motivation to construct this article. Furthermore, those first 64 notations provide the basis for homogeneity and isotropy,  dark energy and dark matter, and a most substantial scale for a unified theory of mathematics which could readily include string theory and the Langlands programs.

5 Comparison to the ΛCDM (Lambda Cold Dark Matter) model. Although there have been many assaults on the ΛCDM, it is the only model that still stands as the parametrization of big bang cosmology. There are seventeen epochs outlined which can be readily and favorably compared within our base-2 timeline. Our initial analysis reinforces the use of the Planck base units as the starting point and base-2  as an initial function for the start and initial expansion of the universe.

6 Natural inflation. We say, “The universe begins most simply before it becomes complex.” Base-2, a simple doubling of the base units, provides a foundational continuity equation that binds the entire universe across the 202 doublings from the first instant to the current time.

This “natural inflationbecame the largest possible continuity equation of the really real.

7 Fundamentality of relationalities. Newton’s absolute space and time are defined as the very nature of existence, always and forever. Relational space-and-time is derivative, discrete, and quantized. What becomes fundamental are the relations that go on forever, that are never-ending and never-repeating  like π (pi) (and probably many of the other dimensionless constants).

8 Four articles extend these foundations:
•  Bottom-up – The simple sphere becomes the foundational  build block for our universe.
•  Map-the-Universe – At the one second mark between notations 143 and 144, the speed of light was mathematically confirmed. To the best our knowledge, this was the first time the speed of light was o9discovered mathematically. Then we found that every notation approximated the speed of light. As we continue our analysis of the numbers, we are confident there are many more similar discoveries that will be uncovered.
•  Transformations –  At the Planck scale we were introduced to cubic-close packing, the Fourier transform, and period doubling bifurcation theory. And with each of these foundational concepts for physics today, we began meeting the greatest scholars within our collective history.
•  Twelve Key Concepts – A steep learning curve, it has been important to try to discern how each concept builds on the next. This article has been written and re-written every year since we began. “Form and function, form and function, form and function” the words of our earliest notions about science, are always being sharpened.

9 Cosmic Microwave Background Radiation (CMBR) Three very fine astrophysicists collaborated to discuss measurements by the European Space Agency’s (ESA) Planck satellite. ESA had been mapping the universe for years and there has been so much data collected, there is more than enough mystery to go around. These three represent an excellent, rather erudite cross-section of today’s research community. Joseph Silk asks foundational questions and is known to be a scholar’s scholar. He would naturally attract like minds and these three were especially bold to conclude that certain results might be suggesting a closed universe.

Joseph Silk, Eleonora Di Valentino (GS, RG) and Alessandro Melchiorri  were especially bold to make this initial analysis of the ESA’s mapping of the CMBR.

Scholars who work primarily within known boundary conditions most often assume their first principles, i.e. the concept of space and time, the initial state of the universe (infinitely hot or not), their essential building blocks (forms), and the initial functions within those building blocks. If they had to define such first principles, they may have also done the same for the ESA dataset and the very nature of cosmic microwave background radiation.

As a result of this article, we will begin a bit more formal analysis of CMBR. Also, just now, I sent a note to a friend saying, “I believe that most of the confusion within astrophysics today is our understanding of space and time. Those 202 notations are not recognized or understood. People do not know from which notation they are measuring values. We’ll help them discern it, by giving them the speed of light at each notation perhaps within a femtometer so when they start to separate out all the redshift more accurately by gauging it against that particular notation’s speed of light (given that most notations have unique light signatures — line 10 in the chart). Yes, yes, yes I am vaguely aware of the far-reaching implications and simplifications of our universe.

10 Little model of the universe. That this model emerged in December 2011 in a high school geometry class may be problematic for many,  but our base-2 exponential model (goes to a little history) actually does encapsulate this entire universe within just 202 notations (goes to the actual numbers).

11.Groundwork: Yes, there are many thought leaders who have trouble with Newton’s absolute space and time. Carlo Rovelli, Richard Muller, and Nima Arkani-Hamed are moving beyond Newton’s absolute space and time. Max Tegmark and Neil Turok have also called for a fundamental re-examination of these concepts. We’ll continue adding to this list as time goes on.

Perhaps we can  call on the two living Lucasian Professors to lead the way to right the wrongs of Lucasian #2 and #17. That would include Professors Mike Cates (#19) and Michael B. Green (#18).

12 Foundations. What can be more fundamental than space and time?  At the outset, the very general statement called “the foundations of our foundations,” was understood to be the precursors of space and time, yet there was no structure for it. Once we the 202 notations were discerned then calculated, those concepts came alive. Now, we can see how each becomes manifest:
1. Continuity, creating order, is the heart and  precursor of time, and an absolute condition for time. Time is derivative, finite and always quantized.
2. Symmetry, creating relations, is the heart of space, a precursor of any actual space, even infinitesimal space. Yes, symmetry is an absolute condition for space which is also derivative, finite, and quantized.
3. Harmony, creating dynamics, is the heart of a space-time moment. It is fleeting, yet embedded within the very fabric of the universe.

These become part of functions within the conversion center between the finite and infinite. So, of course, there will be much more to come!

References & Resources


To be prioritized in order of importance (by the author)

a Philosophiae Naturalis Principia Mathematica (“Mathematical Principles of Natural Philosophy”), Newton, Isaac, author,  London, 1687; Cambridge, 1713; apud Guil. et Joh. Innys, London, 1726  Fellow of Trinity College and the second Lucasian Professor of Mathematics

Leibniz–Clarke correspondence

b The Leibniz–Clarke Correspondence, Edited with an introduction and notes from H.G. Alexander, Manchester University Press, 1956, originally published in 1717. This book is an intimate window on our common history. It is current history. We have to get to know all the players, including Samuel Clarke, Abbè Conti, and Caroline, Prince of Wales. Also, among many online resources, I recommend “Exchange of papers between Leibniz and Clarke, G. W. Leibniz and Samuel Clarke, 1715-1716, Copyright © Jonathan Bennett 2017. Bennett has done a very fine piece of work organizing what amounts to a dialogue about the issues of our time and for all times.

c. Planck evidence for a closed Universe…  There are many crises for cosmology.  This work by Eleonora Di Valentino, Alessandro Melchiorri & Joseph Silk, published in  Nature Astronomy (November 4, 2019) is entirely accessible and readable:  doi:10.1038/s41550-019-0906-9   https://arxiv.org/abs/1911.02087

Let’s more closely examine every crisis within cosmology!


Resources for further exploration

Baryon Acoustic Oscillations (BAO), fluctuations in the density of the visible baryonic matter (the normal matter of the universe), which theoretically are caused by the primordial plasma of the early universe, i.e. acoustic density waves, but within the base-2 model are caused actively today by the very definition of each of the first 64 notations that give rise to the rest of the universe as an ongoing whole. Within the base-2 model it may be argued that the fluctuations are not acoustic, but inherent within certain affects of geometrical configurations.

eefat: Everything, Everywhere, For All Time. A most simple summary of the base-2 model.

General Scalar Fields as QuintessenceA. de la Macorra, G. Piccinelli (1999)


Blackholes: Ripping a hole in space and time, By Kenneth MacDonald
BBC Scotland Special Correspondent, 4 October 2012 

Key words: base-2 model, continuity (order), symmetry (relations) and harmony (dynamics)

More background by Bruce Camber

In 1971, I began work on the foundations of physics by engaging a challenge to define a “moment of perfection in space-time.” In 2011 that work was re-applied to a base-2 exponential definition of the universe from the base units as defined by Max Planck (Nobel, 1918/1919, Berlin). Building on the work of Planck and Frank Wilczek (Nobel, 2004, MIT) to explain the Planck numbers, with our high school geometry classes, we encapsulated the universe — everything, everywhere for all time — within just 202 notations or doublings. With implications for all the sciences, for our understanding of space, time, and infinity, this work is for all people who struggle to understand ultimate things and the finite-infinite relation.  Even more…


Every article within this website prompts emails, tweets, and telephone calls. Here are just a few  with whom we have communicated (initially an alphabetical list with date and link and now added  by date):
1. Prof. Dr. Mike Cates, Lucasian Professor #19, Cambridge University
2. Prof. Dr. Michael M. Green, Lucasian Professor #18, Cambridge University
3. Prof. Dr. Ari Lehto, Finland
4. Prof. Drs. Eleonora Di Valentino, Alessandro Melchiorri & Joseph Silk
5. Then to the following:
John Conway, Princeton
Chris Conselience, Nottingham University
Manjul Bhargava, Princeton
Fabiola Ganotti, Director General, CERN Labs, Geneva
Paul C. W. Davies, Director, Beyond Center for Fundamental Concepts in Science, ASU
Nima Arkani-Hamed, Institute for Advanced Studies
→  Quentin SkinnerBarber Beaumont Professor, Queen Mary University of London
Harry Cliff, CERN and University of Cambridge

Initiated in private on Thursday, November 7, 2019
Very Limited Public Access (No Direct Links): Friday, November 8, 2019
A first draft homepage:  Monday, November 11, 2019
Most active editing: November 7-16, 2019
Last edit of substance: January 25, 2021