Author: Kirk

Reinstalling screws into plastic

We live in a world of plastic consumer goods held together with screws.  These are thread-forming screws with sharp threads that cut threads into unthreaded holes in the molded plastic parts with the intention of never being removed.  Unfortunately sometimes removal is necessary for repair or, increasingly, simply to replace batteries in inexpensive goods.  The problem is that simply screwing the fasteners back in cuts new threads each time, destroying the integrity of the plastic threads and the strength of the joint.  All is not lost, however:

There is a technique to reinstall thread-forming screws without damage.  Place the screw at the start of the hole on the end of your screwdriver.  Using the tips of your fingers loosely on the shaft of the screwdriver, turn it backwards, i.e. counterclockwise, with only the weight of the screwdriver pushing on the screw.  Since the screw is turning backwards the sharp threads will not cut into the plastic.  Turn until you feel or hear a light click – immediately stop.  This means that the screw thread has dropped into the start of the original plastic thread channel.  Now turn the shaft gently forwards, clockwise, to make sure the screw threads slide in smoothly.  If so,  run the screw in.  Do not over torque it.  When you feel it bottom out, stop turning.  That will be firm enough for most plastic assemblies.  If it does not turn easily or feel smooth,  back off and turn backwards feeling for the click and try again.  Some screws are called double lead or hi-lo and have two thread heights.  In that case you will get two clicks, one lighter and one harder.  The harder click is the proper thread channel.  In general it is a good idea to turn backwards at least a full to turn to find the most definite click.  This can also make the best of an already compromised hole.

This technique works as well for reinstalling wood screws without damaging the threads in the wood.  It also works well to avoid cross threading things like wide-mouth cosmetic jars and threaded photographic filters, especially ones made from aluminum.

While I’ve done this for decades, I was reminded that it is not a universally known technique while working on my Dyson vacuum.  These are pricey vacuums with the subassemblies held together internally with screws that you really want to be careful with.  The subassemblies themselves snap to each other.  Dyson only sells subassemblies on their web site that can be replaced without removing and replacing screws.   As all I needed were beater bars and not the entire floor head I kept looking.  I found the smaller parts I needed at evacuum.com for much less than the assembly cost.  After carefully removing the old parts and installing the new screwed on parts without damage I realized why Dyson only sold snap-on parts to consumers but sold smaller parts to dealers.  They did not want consumers to inadvertently  damage their units trying to repair them, but assumed repair shops would know how to do this safely.

Hence this post.  I hope it helps you with future plastic and wood repairs.

Please leave comments using the post in my comments category.

Small Arms Resistant Spotlight

The United States has not fought a symmetric war of front lines and maneuver since Korea.  From then on our actual opponents have not had artillery, tanks, or attack aircraft for more than a few days.  Symmetric warfare has been replaced by asymmetric warfare with the enemy using small arms, IEDs, and infiltration in Southeast Asia, Africa, the Balkans, and the Middle East.  The current response involves fortified bases with cleared fields of fire and targeted operations launched from those bases.  While night vision technology gives US tactical forces a big advantage at night, illumination of the base surroundings makes infiltration harder and aids target identification by troops without NVDs.  The main problem is that conventional spotlights and searchlights are both fire magnets and extremely vulnerable to ordnance.  While economical and very useful for general surveillance, conventional lights can be taken out by snipers at the start of an attack and need to be backed up by something more resistant to small arms that can reasonably survive a short firefight.  Small arms resistance should be sufficient as heavy weapons like artillery, tanks, and attack aircraft will have already been taken care of by the Air Force.  Also it should be possible to keep RPG teams beyond effective range with an adequate field of fire and enough light.  Conventional spotlight weaknesses fall into two categories.  Reflector surfaces are often silvered glass which shatters at the first hit.  A tougher reflector directs ricochets toward the light source, be it a light bulb, arc electrodes, or LED’s which are all easily destroyed by small arms projectiles or shrapnel.  “Bulletproof” glass is not very effective here as high velocity hits will craze it, reducing illumination, and ultimately break through, especially 12.7mm rounds.

The only light source which is truly bulletproof is an illuminating gas flame.  Properly armored reflectors and burners could provide a robust back-up to conventional lighting.  A typical illuminating gas would be acetylene burned in air which produces a bright white light.  Acetylene is readily available and is a normal part of the military supply chain.   Alternately, acetylene could be produced by generators at the light by reacting calcium carbide with a controllable water source.  Other illuminating gas mixtures could be optimized for the application.  Acetylene has been used for some lighthouse lamps from 1896 to the present.

The illustration is for an acetylene tank version with a 3 foot diameter dish.  A carbide version could mount the generators on the back of the dish and eliminate the gas controls except for water shut-offs.

Since any light will be a fire magnet, it is critical that this be remotely operated with perhaps a covered manual override.  The tank is installed below grade or certainly under RPG proof cover.   The idea is to provide four redundant gas channels with separate nozzles, fed through unions in the pitch pivot.  The control modules consisting of the flame arrestor, regulators, and control valves for each channel would reasonably be mounted in the turret and move with it.   Each channel and the main tank would have a zero back pressure shut-off valve so that if a line is severed, flow to that line automatically shuts off.  A possible reflector would be 3/8″ of 17-7 PH stainless steel formed to the desired profile, hardened, and polished to a mirror finish.  This would be bolted to a curved backing plate giving full support made out of armor plate.  The plate would be bolted to the pitch/yaw mechanism.   The burner head would bolt to the mount and contain four separate o-ring sealed passages running between receivers in the mount and the burner nozzles.  Like all parts this would be a plug-in, bolted assembly to facilitate field repairs.  Flame ignition would be electrical or possibly catalytic depending upon gas choice.  A rear mounted chimney of a refractory material would direct the gas exhaust up and away from the mechanism.  This would avoid darkening the reflector with soot and spilling light onto the defenders.  The top of the burner assembly would have a sufficient thickness to resist small arms and have a conical top to deflect ricochets away from the dish surface.  Ricochets from the main part of the dish would pass though the flame without effect.  Near the center, a glacis would deflect shots safely that would otherwise ricochet into the gas nozzles.  The redundant gas channels and auto shutoffs would allow the spotlight to continue operating after losing nozzles.  The illustration is scaled for 7.62×39 but could easily be adapted for heavier weapon resistance.

Body Mass Index

After losing 40 pounds over several years my BMI is now the same as it was when I was in the Army … 50 years ago.  BMI is a formula* used to classify people as underweight, normal, overweight, or obese.  These classes are used as predictors of health, originally intended as guidance for physicians.  When I was in the Army I was in the best shape of my life.  Since I now have the same BMI I should be perfectly healthy and (after 50 years of a sedentary career) should fit into the same clothes.  Of course I do not.  My waist size is still 3 or 4 inches larger and I no longer have the same muscle mass.  While reasonably accurate for mass sedentary populations, BMI treats fat just like muscle and is biased against taller, younger, and athletic people.  Many champion athletes and action movie stars rate as overweight or even obese under the BMI.

*   BMI  =  kg / m²   =  10,000 x kg / cm²  = 703 x lb / in²     Note that this formula requires access to a weight scale, more math than some people are comfortable with, and needs a correction factor depending upon the units used.  Proposed new formulas suggest a 1.3 multiplier and raising height to the 2.5 power, not the kind of thing everyone understands.  Other tweaks include things like multiplying your BMI by your serum albumin level in grams per liter.

BMI is a crock.   It is also an inferior predictor of health outcomes.  This is not news.   The main social problem is that it has evolved from guidance for physicians into a absolute mandate for the bureaucrats who run our children’s lives and is used to bully athletic school children who have more muscle and less fat than their compatriots.

A much better metric is the waist-height ratio.  Measure your waist and divide by your height.  Simple.  All you need is a tape measure.  It doesn’t matter if it’s in inches, centimeters, or old Russian vershoks.  WHtR does a far superior job of accounting for muscle versus fat.  A 2010 study that followed 11,000 subjects for up to eight years concluded that WHtR is a much better measure of the risk of heart attack, stroke or death than the more widely used body mass index.

The US Army has recently switched to WHtR after determining that it is a better metric for body condition.  Even the creators of the BMI maintain that it was only intended for population comparisons, not individual comparisons which it is not suited for.  The question arises, why did they come up with BMI when they knew its shortcomings.  Ah.  They were studying available population records to include health, longevity, and demographic data.  Consider your driver’s license data.  It contains height, weight, and age but NOT your waist size.  Waist size is simply not available in public data bases so they used what was available, height and weight, reasonable for populations but not individuals.

In any case, unlike BMI, waist-height ratio sorts people into a reasonable order:

0.3359   Marilyn Monroe

0.4240   Female college swimmer

0.4280   Male college swimmer

0.4580   Bodybuilder

0.4920   Female at increased risk

0.5000   General healthy cutoff

0.5100   Risk equivalent to BMI of 25

0.5360   Males at increased risk

0.5700   Risk equivalent to BMI of 30

0.5770   Obese

0.5820   Substantial risk increase

As for me, my WHtR tells me I have a ways to go yet.

———————————————————-

The spambots are getting out of hand :  Please leave comments using the post in my comments category.  🙂

Lathe Carriage Indicator

One shortcoming of older small lathes is the lack of a carriage travel calibration unlike on the cross slide.  An X-axis DRO solves this problem but they are not made specifically for older lathes and are often difficult to adapt as the bed and saddle castings did not envision them.  The compound may be used for small, precise X-axis moves but the travel is limited and the compound angle must be indicated in for accurate work.  Older lathes like the Atlas/Craftsman do not have a parallel surface on the compound which makes this quite difficult.  When using the milling attachment which does have a Z-axis dial, there is no alternative as it replaces the compound.  The simplest solution is to use a 1″ or 2″ dial indicator but it is often difficult to mount appropriately using regular indicator holder hardware.  Here is a simple, inexpensive solution:

An adequate magnetic base is under $16 at Amazon or Shars.  The rest of this can probably come from your scrap bin.

This is simply an “L” shaped piece of 1/4″ aluminum with an 8mm hole for the bolt and washer attaching it to the magnetic base and a pinned-in 1/4″ stud to attach and tension the indicator with a nylon lock nut.  The back of the stud is flush with the back of the plate.  In addition to the washer under the nut as shown, there is another washer on the stud under the indicator to provide clearance for the plunger travel past the vertical plate.  Note that the narrow leg is about 1/16″ below the bed so the indicator can sit squarely to be aligned with the carriage travel.  This design works best on flat bed lathes like the Atlas series but can be adapted to others.

The spambots are getting out of hand :  Please leave comments using the post in my comments category. 🙂

Persistence

Nothing in the world can take the place of Persistence.  Talent will not; nothing is more common than unsuccessful men with talent.  Genius will not; unrewarded genius is almost a proverb.  Education will not; the world is full of educated derelicts.  Persistence and determination alone are omnipotent.  The slogan ‘Press On’ has solved and always will solve the problems of the human race. — Calvin Coolidge

Single Malt Scotch

This is not so much a review as a guide to anyone just starting out with single malt Scotch.  A large liquor store may have up to 100 different single malts and in total there have been over 1500 brands which makes a newcomer feel lost.  Unlike bourbon or brandy or a particular wine like sauvignon blanc the various Scotches are completely different.  There is very little useful guidance; most review sites appear to like every kind of Scotch which doesn’t help much in choosing your first couple of bottles.  If you ask random Scotch drinkers they will recommend something like Talisker or Glenrothes or Laphroaig, which you may or may not like, but it won’t help much at the store when you look for the next bottle to try.  I’ve found that choosing at random is a bad idea.

A little quick background will help you understand some of the differences.  For millennia Scotland has been a very poor country.  Most of the Scotch distilleries today started out as illegal bootleg stills in the late 1700’s and early 1800’s with limited access to resources.  Scotch is made from malted barley like beer without the hops.  The barley is moistened to allow it to just germinate (malt) and then is dried with heat to stop the malt.  Here is the first problem; most of the trees in Scotland were cut down hundreds of years ago for firewood.  The usual solution is to dig up peat from various bogs and fens and burn it which dries the barley but also adds often strong flavors that vary by local.  The malted barley is mixed with water in vats and allowed to ferment producing ethanol.  The result is distilled multiple times.  The water comes from a local stream or lake which adds its own flavors.  Most operations were set up near streams with particularly suitable water.  The raw alcohol from the still is not drinkable and needs to be aged in wood casks as is the case for most distilled spirits.  Here is the next problem; new casks were too expensive for the Scots to afford.  The solution was to buy used Kentucky bourbon casks from America or used sherry casks from Spain.

The use of peat, coal, or other fuel for process heat, the type and handling of the Barley, the water source, the details of the stills, the second fill casks used for aging, and even the conditions in the cask house affect the taste of the final product.

By Scottish law Scotch must be aged in the cask for at least 3 years.  As a practical matter commercial Scotch is almost always 8 or more years old.  The “age” refers to the time spent in the wood casks.  Once bottled, the Scotch stops aging.  The older Scotches are not just batches that were left longer in the casks.  They are individual barrels that were identified by the Master of Malt or equivalent at an early stage as especially promising and held for a longer aging cycle.  There may only be a few casks a year good enough to benefit from aging for 18 or 25 years.  A 40 or 50 year old Macallan candidate cask may only be identified once in a decade or so.  The price has nothing to do with taking up space in the warehouse and everything to do with a very limited supply of a great whisky.

The most distinctive discriminant of Scotch flavors is the taste described as smokey, peaty, medicine, or iodine produced by phenols from the peat and possibly sea air.  This taste ranges from very strong in Scotches like Laphroaig to nonexistent in Auchentoshan and Glengoyne which don’t use peat in their process.

By the way, bourbon is whiskey, Scotch is whisky.  Many single malt brands peak in terms of interesting flavor in the 15 to 18 year range.  Younger malts are a little rough and older Scotches seem to pick up more flavors from the cask than from the original process.  For instance I like Macallan 18 ($300) better than Macallan 25 ($1700) and the Macallan 15 ($140) is perfectly nice.  On the other hand while Glenfarclas 17 ($85) is good, Glenfarclas 25 ($160) is wonderful.  Scotch snob warning:  While the Scots usually put a little water in their Scotch, the custom elsewhere is to drink it neat and at room temperature.  If you drink it with ice or even in a chilled glass the cold ruins the taste you paid for.  The Scots also use single malts for cocktails.  It makes me shudder but I suppose that’s what they have.  If you want a Rusty Nail, a Rob Roy, or a Godfather use a blend like Chivas, Johnnie Walker, or Dewar’s and don’t waste your single malt investment.

Like wines in France, Scotland has several Scotch regions with their own typical character.  Unlike French regions where all the white Bordeaux from the Entre-Deux-Mers region is similar, Scotches from each region can vary widely.

Islay is an island west of the mainland and home to Ardbeg, Lagavulin, Laphroaig, Bowmore, and Bruichladdich (bruke-laddee) most of which are strongly peated.  Laphroaig prides itself as “The most richly flavored of all Scotch whiskys” and is very smoky.   Talisker is another highly peated Scotch from the Isle of Skye that reportedly gets its character from the sea environment.

Campbeltown is on the Kintyre peninsula near Islay which once had 30 distilleries.  It is now known for brands like Springbank, Hazelburn, and Glen Scotia.  These are somewhat smoky but less distracting than the Islay malts.

The lowland malts like Auchentoshan are very mild.  It features the tallest stills of any distillery which may account for the smoothness.  Glengoyne 18 is very flavorful in an interesting fruitly way and one of my favorites.  It prides itself as being unhurried since 1833, and “the patience we take over the slowest distillation in Scotland”.  (It’s also the closest distillery to my ancestral home.)

The highland malts are separated into Speysides just south of Moray Firth and the rest of the highlands.  Speysides are very drinkable mostly with limited peat.  Some of the most recognizable brands are Glenlivet, Glenfiddich (-dik), Longmorn, and Macallan.  The Glenfiddich 18 is very smooth while retaining some character.  The Glenlivet verges on bland and since most chain restaurants just carry Glenlivet and Glenfiddich , I always pick the Glenfiddich for the better finish.  Recently Glenlivet moved to perk up their offerings by coming out with a 15 year French Oak Reserve aged in NEW French oak barrels.  This makes a big difference and produces a really rich fruity flavor.  Others have copied this like Balvenie and Glen Garioch (geery) with less successful results.  Macallan has been rightly described as the Roll-Royce of Scotches.  Their products are consistently good with rich flavors, low smoke, and the traditional long finish.  While I usually drink the 15, they have started coming out with annual limited edition versions and the most recent, number 4, is perhaps the best Macallan I’ve tasted yet.  Macallan only uses the first 16% of the ethanol that distills off the barley mash in their whisky.  Distilling for too long produces higher alcohols and other impurities.  Macallan is willing to waste more mash than some for a superior product and is able to charge more to cover the cost.  Longmorn is a good pick but hard to find.  The Scotches I like have a nice aroma and Aberlour A’bunadh has one of the nicest and most intense.  It is bottled at cask strength, 120 proof, and is intended to be mixed 4 parts Scotch to 1 part distilled water to bring it down to 86 proof for drinking.  Don’t cheat here: the distilled water makes a big difference.  It doesn’t work right with bottled drinking water.  The Glenrothes 15 is good but smoky enough to take some getting used to.  A note here is that the way you are feeling, how tired you are, or what you’ve been eating has a surprising effect on how a given Scotch tastes on a given day.  The complexity of some like Glenrothes is more sensitive to this than some others.

Glenfarclas is one of the few privately owned family distilleries in Scotland.  It is currently being operated by the 6th descendant of the original owner.  He is selling Scotch that his grandfather distilled and is distilling Scotch that his grandchildren will be selling.  He takes very great care with what he leaves to them.  The Glenfarclas 17 is tasty; but as I mentioned before, the 25 is wonderful and at $160 it is far more accessible for drinking purposes than the 25 year old Macallan at $1700.

There are over 40 Speyside distilleries that overshadow the 30 or so other highland products but some of the latter are very interesting.  Oban on the Firth of Lorn in the west highlands is good but fairly smoky.  Scapa and Highland Park are only about 5 miles apart on the Orkney Islands north of the Scottish mainland but are completely different.  Scapa is a very smooth Scotch similar to some lowlands brands while Highland Park is more like the heavily peated Islay Scotches.  Macduff which produces Deveron is the easternmost operating distillery in the highlands.  The Deveron 18 is excellent with a nice finish and is not too smoky.  Dalwhinne is the highest distillery in Scotland  —  if nothing else their stills boil at the lowest temperature of any in Scotland.  Their whisky is very smooth even if slightly smoky and is known as “the gentle spirit”.

My current favorite is Glen Morangie 18 (rhymes with orangy).  It has a very nice rich fruity flavor, good finish, and is very smooth.  When I started on single malts, Glen Morangie prided itself as being a small local operation with their whisky “Handcrafted by the 16 men of Tain” for decades.  Tain is a village on the south side of Dornoch Firth about 25 miles north of Inverness.  In the past 30 years Glen Morangie has justifiably become very popular and grown accordingly.  While they still keep the original small distillery running, they have become a major producer and their bottles now say “Perfected by the 16 men of Tain”.  The end product is as good as ever.

Some other good choices: Lismore 18, a very smooth yet inexpensive Speyside whisky, Aberfeldy 18, a richly flavored highland whisky with only the slightest hint of smoke in the finish (their 16 year old is almost as good), and Aberlour 18, another Speyside – perhaps the most richly flavored non-peated scotch, even more so than their A’bunadh. Unlike Aberfeldy, the 18 year Aberlour is much better than the 16 and the price reflects this.

Here is a Wikimedia map of the distilleries in Scotland.

Automated attacks engender wards :  Please leave comments, friend, using the post in my comments category.

Tinactin

If you use Tinactin as a preventative, per the label directions, spray it into the ends of your toes as part of your normal daily application.  If the front corners of your toes are sometimes tender, this will help.  Even though the directions state that the product is not effective on nails, that refers to back under the nails.  It is effective at the edges.

It is best to use the plain Powder Spray (with the grey background) as it is less messy than the Liquid Spray  (with the blue background).  Avoid the Deodorant Powder Spray (with the green background) as the regular Powder Spray works just as well as a deodorant and unlike the regular Powder Spray, the Deodorant Powder spray contains corn starch which is sticky and leaves white marks on clothes, carpets, and shoes.  Speaking of shoes,  Tinactin regular Powder Spray can be sprayed directly into athletic or other shoes before putting them away.  This works to kill shoe odors better than anything else I’ve tried.  Apparently Tinactin also kills the fungi responsible for shoe odor.  Trying this with Deodorant Powder Spray doesn’t work any better and leaves a persistent white residue.

Automated attacks produce automated defenses:  Please leave comments using the post in my comments category.

Spilled Tea

I like Japanese food.  When you ask for hot tea at most of the Japanese restaurants I go to they bring you a cup and a small cast iron teapot with a flat top on the spout.

For a long time, no matter how careful I was, after pouring the first cup there was always a puddle of tea on the table.   Being still more careful made it worse.  I couldn’t even tell where the tea was coming from.  I’ve never been particularly graceful but how hard can it be?  Tip the pot and the tea pours into the cup.

I eventually figured out what was going on.  As the cup filled up I was tilting the pot back gradually, reducing the flow in anticipation of a full cup.  At some point the velocity and inertia of the flowing tea was too small to overcome molecular attraction for the spout and since bottom of the spout ran downhill, the tea ran along the underside of the spout and ran off at the bottom of the pot.  My view was blocked by the teapot so I couldn’t see it happen.

To an engineer this was obviously a design flaw.  The fix was to shape the end of the spout so that tea will always pour out away from the pot and not down the outside of the spout.

Yet many teapot spouts, western and eastern, behave like the first illustration.   Since the time of the Greeks and Romans better spout designs  have appeared on pitchers, beakers,  amphoras and ewers.  Note that on the following pitcher, while the spout does not turn down at rest, it will point down whenever liquid is being poured.

While the flat top teapot spout is bad engineering,  people are good at compensating.  I never noticed until later watching British TV shows on Amazon.  A tea service seems to appear in at least one scene of every British mystery  or police procedural.  British actresses, who have been pouring tea very precisely since they were little girls, automatically snap their wrist back slightly to abruptly cut off the flow as the cup fills, avoiding a spill.

In summary, either (1) this is a major world-wide design flaw requiring mass recalls and government action at the highest levels or (2) since there already is a perfectly good work-around, if you’re not a klutz, it may fall squarely in the Irrelevant Tech category.

Automated attacks produce automated defenses:  Please leave comments using the post in my comments category.

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 it 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:

This 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.