Is Mathematics Discovered or Invented?

This question has been debated by philosophers and mathematicians for at least 2500 years.  Plato maintained that there was an actual plane of existence where “ideal” originals of mathematical objects like circles and squares existed.  Our imperfect renderings of them were shadows of these ideals.  Up until the middle of the nineteenth century, mathematics was the main guide to understanding of our physical world and seemed to be part of it from the mechanics of Archimedes to Kepler’s laws of planetary motion, Newton’s laws of gravity and mechanics to Maxwell’s equations describing electromagnetism.  It seemed obvious that the laws of the universe were mathematical.

Starting in the mid 1800’s Riemann’s non-Euclidean geometry, Cantor’s hierarchy of infinities, and work on the foundations of mathematics were making mathematics more and more abstract and divorced from any connection to the physical world.  By 1920 most mathematicians and many physicists doubted that there was any real connection.  The physicist Richard Feynman famously remarked that if all of mathematics disappeared it would set physics back about a week.

Around this time Einstein’s general relativity was gaining currency and it became obvious that non-Euclidean geometry was actually part of reality.  Shorty thereafter the discovery of quantum mechanics demonstrated that the only possible description of the physics of the microscopic world was inherently mathematical.

In 1960 the physicist Eugene Wigner published an article entitled The Unreasonable Effectiveness of Mathematics in the Natural Sciences discussing the surprising (at the time) fact that some mathematics corresponded so well to the physical world that it could be a guide to discovering new physics and even making empirical predictions.  In a similar vein Richard Hamming wrote: My first real experience in the use of mathematics to predict things in the real world was in connection with the design of atomic bombs during the Second World War. How was it that the numbers we so patiently computed on the primitive relay computers agreed so well with what happened on the first test shot at Almagordo? There were, and could be, no small-scale experiments to check the computations directly. Later experience with guided missiles showed me that this was not an isolated phenomenon – constantly what we predict from the manipulation of mathematical symbols is realized in the real world.

It seems clear that the truth is somewhere in the middle.  Reality has objects – be they atoms, asteroids, or galaxies that can be numbered and compared using integers and rationals numbers.  Geometry corresponds too well to the space and extent of reality to be coincidental.  The non-Euclidean geometries have their place in the Minkowsky space of Special Relativity, the curved space of General Relativity, and in double elliptical form, surveying of the earth.  Group theory captures the symmetries and conservation laws of nature.  Even number theory, celebrated by the mathematician G.H. Hardy as having no practical applications, now is crucial to cryptography, electronic commerce, and even positioning radio telescope dishes.  Analysis and calculus capture the dynamics of the universe.  Any space alien on the other side of the galaxy at least as advanced as Archimedes will know about π and the Fundamental Theorem of Arithmetic.  All this is in the universe to be discovered.

Much of modern and some older mathematics however is purely an invention of the human mind.  Mathematicians have spent a fair amount of time working through the contradictions, conundrums, and paradoxes involved to create a consistent logical framework.  Much of this has been absolutely brilliant work performed by some of the greatest mathematicians who have ever lived.  Keep in mind here that mathematics is not physics and is, at root, simply that which mathematicians get paid to do.  (I only recently learned that there is a polity for this:  If your ideology denies the existence of objective truth, you are forced to maintain that all mathematics is invented.)  Here’s a list of inventions:

(1) Infinity is not part of the universe. It is purely the result of human imagination. In a 1926 paper entitled On the Infinite, David Hilbert wrote “the infinite is nowhere to be found in reality” but he noted that the concept is very useful for some parts of mathematics.  Consensus in the astrophysics community is that our universe is a finite unbounded space, probably a 3-sphere embedded in 4 space, just like our planet is a 2-sphere embedded in 3 space.  As you can travel on the earth’s surface in 2 dimensions forever without reaching an edge, you can travel in space in 3 dimensions forever without reaching an edge.  Our 3-sphere universe is probably embedded in a 4-sphere and so on.  This hierarchy sounds like an infinite universe but not really.  As it turns out the volume for an N-sphere of a given radius only increases up to the 5-Sphere and then starts decreasing rapidly and the infinite sum converges.  If you add up the volumes of all the hyper-spheres of unit radius (say 1 universe radius) the total volume is 45.99932…, ie. the total of volume all the finite universes is finite.  I’m ignoring the various infinite multiverse theories as these appear to be string theorists grasping at straws.

For centuries mathematicians struggled with infinity and the impossibility of doing mathematics with it:  What does ∞/∞ mean?   If 1/∞ = 0 then 0 x ∞ = 1 but it also equals any other number as well.  There was no “natural” way to handle infinity to be discovered.  Mathematicians had to create new mathematics to work with the concept.  In calculus the problem of infinity was solved by the concept of limits.  The following summation equations are now considered to be equivalent but consider the first.

Although this is apparently a standard sum is does not follow the rules for addition.  It is not commutative.  (You can’t add it from right to left.)  By the same token it is not fully associative.

 

Here’s the “commutative” version of the above which clearly doesn’t work:

The second summation, while “equivalent” is fully commutative and associative.  Modern mathematicians understand that this is what is meant by the first version but it was really confusing 400 – 500 years ago.

The problem is trying to use infinity as something you can reach or a value that can be assigned to a variable.   It is not obvious that the mind can truly imagine infinity (as opposed to a really, really big number).  Infinity can be used as a direction, or “and so on”, or a limit but trying to use it as an actual number or pretending you can get there, while a part of mathematics, puts you firmly in Wonderland in terms of reality.

(2) Self-reference is not part of the physical universe.  Most if not all paradoxes rely on self-reference for their action.  Observe a trivial example:

The next sentence is true.  The previous sentence is false.

Clearly if the first sentence is true, it is false – a paradox.  Much of modern mathematics also relies on self-reference such as Gödel’s Theorem or the modern definition of a set.  Russel’s Paradox was created at the dawn of set theory by self-referential sets.

The universe doesn’t appear to contain self-reference or even abstraction unless you want to count the holographic principle which is more like two views of the same object.  Going back to the simple example earlier, self-reference, paradoxical or not, involves a  flow or motion that circles back on itself.  First there is the logical evolution through logical statements to the conclusion that turns back on itself and potentially modifies part of the past.  Wherever you have self-reference there is the danger of a circular paradox.  The universe apparently eliminates any chance of this by freezing the past and outlawing time travel.  In the invented part of mathematics self-reference is just fine – as long as you are really careful.  Bertrand Russell and others spent years patching the self-referential holes in set theory.

(3) Axiomatic systems are obviously invented.  Our space alien friend is clearly not going to “discover” our axioms.  A philosophical problem with sets of axioms is the hubris that leads a person to assume that they can, apriori, capture all the truths and relations of a branch of mathematics in a short set of statements.   This is generally not even true.  Gödel demonstrated the problem with something as simple as arithmetic.  Euclid’s fifth postulate was argued over for 2400 years and eventually subsumed in modern times.  Even the most carefully crafted modern set theory axioms cannot capture or define a particular mathematical system, and according to the Löenheim-Skolem Theorem, they can be satisfied by completely contradictory mathematical structures.  Axioms are like the rules of a game such as chess or go and can lead to vastly interesting results.  Axioms can also be like the rules of golf or baseball and can lead to all kinds of contentions.  In any case they are, in a sense, arbitrary.

(4) Sets are totally a product of human imagination.  These are relationships among objects, real or imagined, that are absent from the universe.  About 130 years ago mathematicians got the idea that an obscure object in combinatorics, the set, could be used to patch up the logical foundations of mathematics which were tottering.  It was a classic case of the tail wagging the dog.  Trying to stretch sets to encompass all of mathematics resulted in paradoxes and tsuris leading to “solutions” like Russell’s type theory which is strongly reminiscent of the epicycle theories used to patch up the earth centered Ptolemaic solar system models.  Axiomatic set theories such as Zermelo-Frankel attempted to bring order to the ensuing chaos with limited success.  The ultimate result was a whole spectrum of axiomatic set theories, all different, and all resulting in a different mathematical structure, kind of like with and without a designated hitter, only worse.  Adding axioms to Z-F like the axiom of choice or the continuum hypothesis lead to things like Banach-Tarski or non-measurable sets.  The last has a certain historical poignancy.  Classically a standard method of proof since Pythagoras’ time was proof by contradiction:  You assume the inverse of ‘A’ and then if a logical progression leads to a contradiction, ‘A’ must be true.  By contrast, in the case of non-measurable sets, a contradiction simply leads to the invention of a new kind of set that can’t be measured.

(5) This is a bit of a pedagogical issue but the problem has existed ever since Euclid’s first axiom defined points.  To be perfectly clear, there are no points in nature.  There are locations, addresses, coordinates – but there’s nothing there.  Treating a point as a real object caused all kinds of problems for thousands of years as mathematicians, working to calculate lengths and areas before calculus, tried to figure out how to add up an infinite number of zero sized points to get a given finite line length since a line was made of “points”.

To sum up: Reality pretty clearly runs on mathematical principles and is almost certainly consistent.  Reality and math are part of the same fabric even if we sometimes have a little trouble figuring out those principles.  There is also a lot of mathematics that has nothing to do with the universe.  Peter Woit, a theoretical physicist, believes that this conflict exists in string theory where very abstract models may be impossible to test in any foreseeable experiment.  If this is the case, the “string” must be thought of either as real but untestable, or simply as an illusion or artifact of either mathematics or cognition.  Finally, if using Zermelo-Frankel set theory axioms with the “axiom of choice” added, you have infinite point sets that allow you, per Banach-Tarski, to disassemble a sphere into five pieces and reassemble the five pieces into a sphere one half the size of the original, (a provable result under ZFC) you are so far down the rabbit hole that Wonderland disappeared in the mirror long, long ago.

Appendix Carry

For decades there have been four basic, agreed upon, civilian firearms safety rules.  (Police and military are slightly different.)

  1. All guns are always loaded.
  2. Never point the muzzle at anything you are not willing to destroy.
  3. Never touch the trigger until the sights are aligned with the target.
  4. Be sure of your target and what is behind it.

I have been noticing the appearance of so-called appendix carry including an inside the waist band variant, AIWB.  This places the holstered handgun in front of the strong side hip against the tummy, and for right handers, over the appendix, hence the name.  I suppose the idea is a quick draw from under an untucked shirt.

This is a really bad idea and seriously violates rule #2.  Expecting to use this for a fast draw under combat stress is a worse idea.  The muzzle is pointing directly at you-know-what if you’re male and similarly valuable parts if you’re female.  When sitting down it is pointed at your femoral artery.  These certainly fit in the not to be destroyed category.

All the traditional carries, back of the hip strong side, SOB, cross draw, western, and various shoulder rigs do not point the handgun at the wearer for a reason.

If you are new to concealed carry, be aware appendix carry is a dangerous fad to be avoided.   Massad Ayoob‘s StressFire explains how badly combat stress interferes with fine motor control including trigger discipline.  It is written for police training and tries to convey the difference between a range session and a firefight – kind of like the difference between seeing a picture of a roller coaster and riding one.  An excellent resource is Ayoob’s In The Gravest Extreme: The Role of the Firearm in Personal Protection.  This is written for civilians and ranges from tactics to legal issues.  Massad Ayoob is a highly respected police trainer, self defense author, and founder of the Lethal Force Institute.   Another resource is Jeff Cooper’s Principles of Personal Defense.  Arguably the best books written on the reality and effect of lethal action are Lt. Col. Dave Grossman’s On Combat and On Killing.  The point of reading these is to try to understand what’s going to happen in the event.

Draining Garden Hoses

If you drain garden hoses in the fall or between jobs and have a portable air compressor you may find this adapter a big time saver.  It is assembled out of a few dollars worth of parts from the hardware store:  a ball type hose valve, an air hose adapter, an appropriate thread adapter, and thread tape.

With the valve off, attach the adapter to your air compressor.   Run the compressor to shut-off, unplug it, and carry it to your hose.  Remove the hose nozzle and secure the hose end as it may thrash around.  Disconnect the hose from the water source and attach it to the adapter.  Crack the valve open slightly to start the water flowing.  Moderate the valve to get a typical flow.  When the hose starts to sputter, briefly open the valve fully to blow the last of the water out.

This will empty reeled hose as well as coiled or loose.  A 3 gallon tank will clear a 50 foot hose in one shot.  A 6 gallon tank is sufficient for a 100 foot hose or a couple of 50 foot hoses.

Chopped PID Control of Processes with Delay

For nearly 100 years PID controllers have been the standard for feedback control of a wide range of processes including autopilots, automotive cruise control, industrial heating, servo positioning, and motor speed control.   PID stands for Proportional-Integral-Derivative, also called Span-Reset-Rate in earlier times.  The PID controller reads the process variable, PV, such as temperature or position to be controlled.  The value is compared to a setpoint, SP, and use to calculate a control variable, CV, such as heater power or motor voltage to drive the process toward the setpoint.  PID controllers are “tuned” by adjusting three internal constants.  Ideally the tuning should allow close control of the process.  A properly tuned controller should bring the process up to the desired setpoint quickly and settle smoothly to the desired value without overshoot.  A controller that is too aggressively tuned will slew the process too hard and overshoot, then try to bring that process back, undershooting, and continue to oscillate around the setpoint.  A controller that is too conservatively tuned will react sluggishly and slowly approach the desired value, perhaps never reaching it.

PID works well for ordinary processes but has a problem controlling processes with a significant delay compared to the process response time constant.  An exotic example would be a remotely controlled lunar rover.  It may be able to turn in 1 second but nothing will be seen for 3 seconds and any error will take 3 seconds to change.  Imagine a simple earth based direction control where the rover hunts back and forth around the desired direction, tacking one way for 3 seconds and then the other for 3 seconds.  If a standard PID controller is tuned conservatively enough to eliminate the hunting it will be very sluggish compared to the potential speed of the overall controlled process.  A more familiar example is adjusting a bathroom shower.  If it’s too cold you turn down the cold faucet, but nothing happens for few seconds.  If you keep turning the cold down while waiting the water will end up too hot.  Then you start turning the cold back up and when the water at the shower head hits the right temperature it is already too cold coming up the pipe.  The problem here is that the temperature can be changed much faster than you can find out about it.  A human quickly hits upon the solution:  make a small adjustment, wait a bit, make another small adjustment, wait, and repeat until the temperature is correct.

Modern computerized process control uses strategies such as the Smith predictor which uses a mathematical model of the process to predict in advance what the process will do after the delay or System Inversion which uses a model of both the process and the controller for an improved future prediction.  These require a priori mathematical computer models of the process and controller.

Long before such sophistication existed there was a surprisingly robust technique called chopped or pulsed PID.  This basically emulates the human shower algorithm by chopping the PID controller on and off repeatedly.   This is still useful when a simpler solution is desired or when there is limited knowledge of the process model, as is usually the case.

The PID is first turned on briefly.  While ON the PID operates normally, driving the process toward the desired result.

The PID control is then turned off for a time approximating the process delay.  While OFF the proportional and derivative terms sample the error normally but have no other effect beyond keeping track of the rate of change.  If the integral gain is nonzero, the internal integral term is held unchanged.  If the process is integrating such as position control, the control variable is turned off (zeroed) while the PID is off.  If the process is a normal lag such as temperature or speed control, the control variable is held unchanged while the PID is off.

This adds two more constants to the existing three for PID.  The off time should approximate the delay, longer is more conservative, shorter is more aggressive.  An on time less than the process time constant exclusive of delay is more conservative and longer is less conservative.  Neither is particularly critical and the control scheme is quite robust.  When tuned properly, chopped PID responds and settles much more quickly and stably than standard PID for a process with delay.

Processes that involve pure delay include: fluid flow in pipes where the source is heated or cooled but the pipe outlet temperature is sensed, speed of light delay for control of satellites, pneumatic instrumentation and control systems where small pressure changes have to propagate through long thin tubes, and control over computer networks where there can be significant processing and/or communication delays.

Pruning Loppers

This post addresses pruning loppers that are two or three feet long but also applies somewhat to one hand pruners and pole pruners.

The basic rule for using a lopper is: NEVER twist the lopper.  If the lopper does not cleanly cut all the way through a branch and sticks, rotate it back and forth around the branch in the plane of the cut to get it unstuck.  It is tempting to give it a little twist to snap off the branch or pry it out — resist that temptation.  If not you will often chip the cutting blade and ruin it.  The cutting blade of a lopper is very narrow and very hard in order to make it cut easily.  It is immensely strong in the direction of the cut but quite brittle against side loads, which is what makes twisting or prying so risky.

There are three basic types of loppers.  The first is the anvil style that pinches the branch with a sharp cutter against an anvil jaw with a brass or polymer insert to keep from dulling the cutter.  The cutter and anvil are usually straight.  The main problem is that if you try to cut a green branch of even moderate size it tends to squirt out of the jaws.  These also often take a greater amount of effort than other designs.

The second type is a “Bird’s Head” bypass lopper.  In these, the curved cutting blade “bypasses” the fixed blade like a pair of scissors.  This usually results in lower cutting effort for a given design.  But, the curve of the cutting blade matches the curve of the fixed blade. As a result green branches tend to squirt out of this style too.

The third type is a bypass lopper with a cutting blade shape that is more squared off than that of the fixed blade.  This captures the branch to be cut, eliminating any possibility of it squirting out.  As a bypass lopper it cuts easily and the captured branch is held in an optimum position for the cut.  Several manufacturers make these.  The Fiskars PowerGear® and Power-Lever® families are good examples of the type.

A major consideration is replacement parts.  The part that generally wears out, even with the best of care, is the cutting blade and a worn or chipped cutter is an exercise in frustration.  If your lopper has replacements available they will be much cheaper than buying a new lopper.  Some manufacturers sell repair parts on their web sites.  As an example, a search for “lopper parts” on the Fiskars web site returns various cutter blades for current and past loppers.  (It also lists a number of lower cost loppers that are still in production.)  Drilling down into likely pictures and descriptions yields model numbers and reference dimensions so you can be sure you are ordering the right part.  A replacement cutter is $8 for a $46 lopper and $4 for a $25 lopper, much better than a whole new lopper from the hardware store.  Buy a couple while you’re at it.

454 Casull Penetration

The 454 Casull was originally developed as a high performance hand gun hunting cartridge.  With about 50% more energy than a 44 Magnum it has become very popular as a bear defense handgun cartridge.  When in bear country, make noise while walking, use your tiny bells, and call out ‘hey bear’ or something periodically – Plan A.  Wild animals will often shy away from anything unusual.  If you see a bear heading your way with intent, a cloud of bear spray in front of the bear will often discourage it – Plan B.  If not, remember, bears kill and eat large animals on a regular basis.  That is nature, red in tooth and claw.  If you don’t feel like being  part of the nature menu right now it’s time for Plan C.

The main problem is that a bear has hundreds of pounds of massive, thick skull, thick fur, tough hide, heavy dense bones, gristle, and muscle protecting its vitals.  Ordinary handguns won’t do much to a Brown bear except make it mad.  As for all dangerous game, penetration is the most important consideration.  A flat meplat or bullet nose helps.  In preparation for a Western hiking trip I decided to test various loads both straight on and at a 45° angle to simulate a glancing impact.  The target was 1.5” of Purpleheart to simulate a skull or heavy bone, backed up by 6” of spruce in the form of four 2×6’s all held together by staggered 3” deck screws for a total thickness of 7.5”.  Purpleheart is the hardest wood I had readily available.  It is considerably harder and more crush resistant than Rock Maple, for instance.  For the 45° angled shots the slant thickness was 10.5” on the diagonal.

Ruger Super Redhawk, 454 Casull, 7.5” barrel, 21 feet from target

90° penetration

90° retained weight 45° penetration 45° retained weight

Buffalo Bore 7B 300gr JFN (exposed lead nose) 1550 fps

3.7”

284gr

3.0” (a)

288gr

Corbon Hunter 335gr HC (Hard Cast) 1550 fps

4.6”

194gr

4.6”

258gr

Corbon Hunter 325gr FPPN (Flat Point Penetrator) 1550 fps

> 7.5” (b)

325gr

10.1”

325gr

Federal Swift A-Frame (hollow point) 300gr 1520fps

5.0”

268gr

6.7” (c)

279gr

Kimber 1911 45ACP, 5” barrel, 21 feet from target

Remington UMC 230gr FMJ 835fps

2.1”

230gr

1.8” (d)

230gr

Remington Golden Saber +P 185gr JHP 1140fps

3.0”

185gr (e)

2.6”

185gr

Notes:

(a) The corner of the lead tipped JFN apparently caught on the 45° impact. It tumbled sideways and didn’t penetrate as far as the 90° shot.

(b) The 90° Corbon FPPN blew straight through the 7.5″ stack, the only round that did so.  Even though the bullet was not recovered there was no jacket or core material in the channel through the wood.

(c) The 90° Federal Swift A-Frame expanded as expected, retaining 89% of its weight.  The 45° impact shot collapsed the A-Frame hollow point from the side, causing  that round to not expand significantly, to penetrate farther, and to retain more of its weight.

(d) The 45ACP FMJ distorted for the 45° impact and tumbled slightly resulting in less penetration.

(e) Both of the 45ACP Golden Sabers packed Purpleheart into their hollow points and did not expand.  They also did not loose any weight.  The 45° shot tumbled slightly.  I added the 45ACP to illustrate the difference between a personal protection cartridge and a bear cartridge.

Note that the 45° Corbon FPPN penetrated 10.1″.  It kept straight all the way and was not diverted or destabilized by the angled impact.  This is a hard lead core flat point bullet with a thick jacket that completely covers the core including the base.  Here is what one looks like after going through 10.1″ of wood.

Unfortunately it’s not available anymore.  If you are looking for this kind of penetration there are some alternatives.  Grizzly sells a 300gr 1400fps metal jacket lead core round.  These use the Belt Mountain Punch bullets.

Another possibility is the Magtech 260gr 1800fps lead core FMJ Flat.

There are monolithic copper penetrators from manufacturers such as Underwood.  As they are less dense than cored bullets, they take up more powder space for a given bullet weight and can’t be loaded to the same energy levels as a lead core bullet.

I’m waiting for someone to make a tungsten cored heavy brass jacketed flat nose penetrator.  As tungsten is denser than lead, this would leave even more powder space for a given bullet weight.

One common recommendation for this application is the Hard Cast bullet.  Note that these penetrated less than half as deeply as the FPPN with a heavier bullet at the same muzzle velocity.  Here’s the 90° HC after 4.6″.  Also note that it only retained 58% of its weight.

The Federal Swift A-Frame penetrated deeper than the hard cast, retained over 89% of its weight, and expanded to 0.75″ as recovered.

As it turns out, I have enough Corbon FPPN for all the predatory bears I could conceivably meet so that’s what I’ll be carrying.  If I couldn’t use that,  I’d either take Federal Swift A-Frames or repeat the test using Magtech FMJ Flat, Grizzly Punch, and the Underwood Penetrators.

Quad stopper knot

This is a stopper knot that is somewhat bulkier than Ashley’s Stopper Knot and is derived from it.  Where Ashley’s shows a trefoil pattern when looking down the standing end, this knot has a four-fold pattern around the  standing end.

Topologically the knot is a double overhand noose with a tuck back.

It is easiest to tie without the tuck at first.  Tighten up the double overhand knot around the standing end by eliminating the A segment.  This will bring the B and C legs together with the standing end inside the loop.  Snug this up enough that the standing end is no longer free to slide and none of the sub-loops are loose.  The tighter this is made, the more secure the finished stopper.

Then, continuing in the same wrap direction around the B segment, tuck the working end through the noose as shown.  Pull the standing end and the working end tight to create the four-fold pattern.  In addition to being bulkier, if this knot comes loose there is a remaining overhand knot as a backup stopper.

If this knot already exists, please let me know so I can give proper credit.

Old-school combs

A long time ago, when consumer products were made in America, the original ACE combs were ubiquitous.  Men and boys carried them.  Every barber shop had an ACE comb sales display.  James Dean famously used one to comb his hair in Rebel Without a Cause.  President John F. Kennedy owned at least one.  ACE combs were successful because they were quality products, typical for that age.  They were made from a tough, strong, flexible hard rubber with thick end guard tines to protect the teeth.  The rough mold parting line around the back and edges was carefully ground off smooth to eliminate the sharp edge.  Even the tips of the teeth were smoothed off so as to not damage hair. They could be dropped without damage and tended to last for years.  As cheaper imitators appeared, ACE started stamping their products as “Genuine ACE Hard Rubber”.

During the adverse conditions of the 1980’s, one of their competitors bought ACE and that was the end of the original ACE comb.  Since the competitor now owned the ACE trademarks, they had legal right to stamp “Genuine ACE Hard Rubber” on their existing combs made out of a more brittle material with thinner guards and an exposed sharp mold line including along the tips of the teeth.  These combs broke if you dropped them on their ends but that just meant the customer would buy another.

Many people didn’t notice, but if you cared, it was annoying.   If you do care, here are a couple of sources for hand-made, high quality combs, with saw cut teeth to eliminate mold edges between teeth, and polished all over so they have no sharp edges anywhere.  They are tough and long lasting.

Speert imports a wide range of private label high quality hand-made Swiss combs for men and women, in pocket, purse, and styling versions.  The Speert site also offers a very wide selection of different styles, sizes, and diopters of inexpensive reading glasses.

https://speert.com/combs/

For seriously old-school, Kent has been making the “world’s finest brushes” and combs for over 240 years … since 1777.  They produce a range of men’s and women’s hand-made, saw cut, polished combs.  You can hear the difference as their combs glide through your hair.  The given link is for Great Britain and while they have importers, the home web site is worth visiting.  It has a web store and PayPal does the pound/dollar conversion seamlessly.

https://kentbrushes.com/

These are elegant combs for a very reasonable price.

New M1 Garand Ammunition

Do not ever shoot regular 30-06 hunting ammunition in a standard M1 Garand.  Commercial  30-06 ammunition is loaded to a much higher pressure than the “Cal .30 Ball M2” that the Garand gas system was designed for and simply will not work.     At best, commercial ammo will cause a stuck case and at worst will bend the operating rod and/or break the extractor.

The last volume source for M2 Ball equivalent military surplus ammo was Greek HXP which was happily non-corrosive, unlike WW II and Korea vintage U.S. military surplus.

As surplus HXP started to dry up Federal came out with 30-06 ammunition loaded to Garand specifications.  The boxes were printed with a picture of a Garand and the part number had an M1 suffix.  Since then two other manufacturers have realized that with 6 million Garands floating around and Fulton Armory making new ones there is a significant market.  The three current sources are:

Federal /American Eagle AE3006M1

Sellier & Bellot SB3006M2

Prvi Partizan PP347

A web search will turn up multiple ecommerce sites for each.

If you’ve never shot an M1, don’t pass up an opportunity.  There are few firearms as much fun to shoot as the Garand.