Mathematics to many people is completely abstract and thus, per definition, is not doing anything physically. But is that true for math in a processor? In this article I will describe how mathematics in computers is applied to actually “do something.”
It is well known that mathematics is used to model physical phenomena. We all know the formula E=mc2, which says that an energy equivalent to a mass m is equal to the product of the mass m and the square of the speed of light. You can calculate the energy of a mass, but the formula does not tell you how to release that energy. And calculating the energy by executing E=m*c2 for 1 gram does not release that energy. That supports the “abstract” part of mathematical expressions.
Another model of a physical phenomenon is Ohm’s law: V=I*R, which says that the voltage over a resistor of R ohm is the product of the current I through the resistor and the resistance R of the resistor. Executing this formula in your mind or on a computer does not generate a voltage. It would also be absurd to assert that the resistor performs a multiplication. That is not is not the meaning of Ohm’s law.
Movie as sampled reality
A movie is a rapid display of individual pictures at a certain rate, like at minimum 12 images per second. The human vision system works like a sample and hold system and at a certain display rate our brain can no longer distinguish individual images and sees a moving picture. A movie or video camera samples the visible spectrum at a predetermined rate and a movie is a sampled representation of reality. The images of a film movie are printed on the film, previously celluloid and nowadays a polyester film. The film movie is a material thing with analog samples (images) in sequence placed on the carrier which is spooled on a reel. Images in a digital camera are data elements obtained from a light intensity sensor rather than a physical print on a carrier. The signals from the sensor elements are analog and are quantized by an A/D converter into data elements. From that moment forward, until the image data is used to activate a display device, no images as recognized by a human exist. Raw image data from image sensors have several inherent errors that have to be corrected or minimized by processing of data. One operation performed on a digital camera is spatial interpolation between pixels, which is performed in accordance with a mathematical formula. This is truly mathematics based and no physical equivalent exists. We can explain what the physical effect is of interpolation and what would happen if no pixel interpolation was applied. However, the physics only come into it after D/A conversion of data. Up to that moment there is only data processing, mostly in accordance with mathematical expressions.
The Sampling Theorem
The basis of many digital devices is the sampling theorem. It states that when signal samples are generated at a speed of at least 2 times the highest frequency component in the signal, then the original signal can be fully reconstructed from the samples. And apologies to anyone who finds the above trivial. However, some people have picked up some terms and concepts but draw the wrong conclusions about computers and math.
Applying the sampling theorem involves creating a sample of a signal, digitizing it into a word of bits, rapidly processing the word of bits in accordance with a processing rule into a new word of bits, converting the new word of bits into an output signal that may be used to control a screen or earphones on a timely basis. This is fundamentally how your smartphone works. The processing rules are mathematical formulas in accordance with which physical switching circuits are created and activated. (There is nothing “abstract” in that step. But that is not the point of this article)
One example of a digital sampled device is a low-pass filter realized as a moving average filter. The working is simple. A device receives a signal and samples it at let’s say hundred times per second. A processor for instance takes the last 50 samples after a new sample has been received and calculates the average of these samples and generates a signal from the average and outputs that average as a “filtered” signal. The mathematical averaging diminishes the effect of spikes in the sample value. Spikes may be considered to be higher frequency signals. Thus the mathematics establishes a low-pass filter.
For real-time operations it is required that the instructions that operate in accordance with the mathematical expressions generate the output samples on a timely basis without delay. With GHz clock-speed processors that is generally not a problem. These devices are in essence calculators with additional functionality/devices: a memory to store data, a clock mechanism to process data at the right moment, a delay and feedback line usually implemented as a shift register. What makes a sampled calculating device a filter or a modulator are the A/D and D/A converters that change a signal into a data element and converts a data element back into a signal.
Filters and modulators used to be devices made from resistors, inductors, capacitors, amplifiers and a lot of connecting copper. These are now replaced by instructions determined by mathematical expressions that are cleverly implemented on a computing machine. You really have to let this sink in. The math no longer is an abstract model of reality, it is reality. We are getting to a stage where it may no longer matter if you have data or a physical expression of the data. For instance a modification of a data file that can be expressed by a 3D printer into an object is for all practical purposes a machine that works on the object. Here are some other well established examples:
Many medical images such as MRI images do not really exist in the sense of a photograph which captures light on a light sensitive medium. Signals generated in MRI are artificially created and captured by sensors and converted into data. After data are obtained, mathematical expressions are used to extract a data slice that is displayed as an image on a screen. Sometimes it is desirable to display an image of organ, which exists in different slices. The technique to extract a 3D image of an organ from a data volume is called 3D rendering and is basically a series of mathematical operations on data elements or voxels. Until an image is displayed on a screen, the processor in an MRI machine is a big calculator operating on data.
A Global Positioning System (GPS) works off signals received from satellites. To calculate a GPS position and to minimize errors a set of equations is solved based on data derived from the received signals. While receiving satellite signals is not a trivial operation, it is fair to say that a significant part of a GPS device is a calculating machine.
Many control systems either in cars, airplanes, thermostats, toilet flush systems, power generating systems work with feedback. A pre-determined setting is checked against a state of a device and automatic control activates a mechanism or device based on an error between the pre-determined setting and actual state. Rather than trying to build a control system in electronic components, it is often easier to model the system in a set of equations (state-space equations for instance) and program compensating equations in a processor. Control systems, except for its sensors and actuators, include mainly a programmed calculator.
Public key cryptography is one of the most widely used security applications. At the heart of public key cryptography are truly hardcore mathematical operations. One mathematical operation that is often used is not more complicated than high school mathematics: modular exponentiation or repeated modular multiplication. The other operation that is used is a bit more involved and is called elliptic curve cryptography. But it is all math that is used to provide instructions to a computing device.
Many, many more examples can be given of the use of mathematical expressions in useful devices.
Mathematics has long been accepted as a tool to model the physical reality. For many it is hard to grasp that math actually “does something.” The reality is that mathematics based instructions in computers generate signals that are useful and used. This type of mathematics may be called “operational mathematics.” Operational math already replaces devices that used to be made from valves and gears or from electronic components. Operational mathematics also enables new devices that were previously unimaginable.
Mathematical expressions continue to trigger 35 U.S.C. 101 rejections in patent applications. These rejections are scientifically arbitrary actions as claims are being allowed that use terms like: Fourier Transform, correlation, wavelet, multiplicative inverse, polynomial, finite field, convex optimization, multiplication, exponentiation, Bayesian, estimation and even computing. These are all mathematical operations that correspond to mathematical expressions.
Operational math on a processor is a switching device and not an abstract idea. However, issues of operational math go beyond patent eligibility. Due to its complexity it is often unrecognized by the general public and considered to be “just math.” I am pretty sure that most Examiners recognize “operational math” for what it is, but are instructed to ignore what they learned in school. We are at the beginning of a massive use of “operational math,” of which many further examples can be provided. The coded world is rapidly becoming the real world, while some people appear to still see a mechanical world described by abstract math. That mechanical world has passed and the role of “operational math” is becoming more influential than ever.