ΑΠΟΨΕΙΣ ΓΙΑ ΤΑ ΜΑΘΗΜΑΤΙΚΑ



The back of a tiger could have been a blank canvas. Instead, nature painted the big cat with parallel stripes, evenly spaced and perpendicular to the spine. Scientists don't know exactly how stripes develop, but since the 1950s, mathematicians have been modeling possible scenarios. Now researchers assemble a range of these models into a single equation to identify what variables control stripe formation in living things.

This image shows simulations of Turing stripes. On the left, stripes are evenly spaced, but their direction is variable. On the right, a signaling gradient has made the stripes align in the same direction.
Credit: Tom Hiscock
The back of a tiger could have been a blank canvas. Instead, nature painted the big cat with parallel stripes, evenly spaced and perpendicular to the spine. Scientists don't know exactly how stripes develop, but since the 1950s, mathematicians have been modeling possible scenarios. In Cell Systems on December 23, Harvard researchers assemble a range of these models into a single equation to identify what variables control stripe formation in living things.

"We wanted a very simple model in hopes that it would be big picture enough to include all of these different explanations," says lead author Tom Hiscock, a PhD student in Sean Megason's systems biology lab at Harvard Medical School. "We now get to ask what is common among molecular, cellular, and mechanical hypotheses for how living things orient the directions of stripes, which can then tell you what kinds of experiments will (or won't) distinguish between them."

Stripes are surprisingly simple to model mathematically (and much of the early work on the subject was by Alan Turing of "The Imitation Game" fame). These patterns emerge when interacting substances create waves of high and low concentrations of, for example, a pigment, chemical, or type of cell. What Turing's model doesn't explain is how stripes orient themselves in one particular direction.

Hiscock's investigation focused on orientation--e.g., why tiger stripes are perpendicular to its body while zebrafish stripes are horizontal. One surprise from his integrated model is that it takes only a small change to the model to switch whether the stripes are vertical or horizontal. What we don't know is how this translates to living things--so, for a tiger, what is the variable that pushes the development of perpendicular stripes?

"We can describe what happens in stripe formation using this simple mathematical equation, but I don't think we know the nitty-gritty details of exactly what molecules or cells are mapping the formation of stripes," Hiscock says. Genetic mutants exist that can't form stripes or make spots instead, such as in zebrafish, but "the problem is you have a big network of interactions, and so any number of parameters can change the pattern," he adds.

His master model predicts three main perturbations that can affect how stripes orient: one is a change in "production gradient," which would be a substance that amplifies stripe pattern density; second is a change in "parameter gradient," a substance that changes one of the parameters involved in forming the stripe; and the last is a physical change in the direction of the molecular, cellular, or mechanical origin of the stripe.

Although this paper is based in theory, Hiscock believes that we are close to having the experimental tools that can decipher whether the math holds true in living systems.

Τα μαθηματικά αντικείμενα  συνίστανται από καθαρή σκέψη…. Η σχέση ανάμεσα στα μαθηματικά και την πραγματικότητα δεν είναι προφανής… H πραγματικότητα του χρόνου επηρεάζει σημαντικώς  τον ρόλο των μαθηματικών στη φυσική…Κάποιες πτυχές του πραγματικού σύμπαντος δεν θ’ αποδοθούν ποτέ με μαθηματικό τρόπο….η φύση δεν μπορεί να αιχμαλωτισθεί από κανένα λογικό ή μαθηματικό σύστημα…τα μαθηματικά δεν αντικατοπτρίζουν  όλες τις  ιδιότητες της φύσης…Τα μαθηματικά  δεν αντιστοιχούν στις πραγματικές φυσικές διαδικασίες,αλλά απλώς τις καταγράφουν…στην πραγματικότητητα τα μαθηματικά έπονται της φύσης…είναι ανόητη η άποψη ότι τα μαθηματικά προηγούνται της φύσης…Τα μαθηματικά είναι μια γλώσσα της επιστήμης…τα Μαθηματικά θα συνεχίσουν να είναι η θεραπαινίδα της επιστήμης,αλλά δεν μπορούν πλέον να είναι η Βασίλισσά τους…
Lee Smolin (“ O Χρόνος»)

Οι αλήθειες του μαθηματικού κόσμου υπάρχουν σε μια πραγματικότητα που δεν συλλαμβάνεται από κανένα σύστημα αξιωμάτων.
Roger Penrose

As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain they  do not refer to reality
Einstein
Αλλά
How is it possible that mathematics,a product of human thought that is independent of experience,fits so excellently the objects of physical reality?
Einstein

Μπέρτραν Ράσσελ., ο οποίος, αναφερόμενος; στα καθαρά μαθηματικά έλεγε ότι « αυτά είναι κάτι για τα οποία δεν ξέρουμε γιατί μιλάμε και ότι αυτό για το οποίο μιλάμε,δεν είναι αλήθεια»
-Ο Godel έλεγε ότι τα μαθηματικά, αντικείμενα υπάρχουν μακρυά σε μια άλλη διάσταση,ότι ασχολούνταν με την αντικειμενική ύπαρξη των μαθηματικών  αντικειμένων.,ότι ο χρόνος δεν υπάρχει υπό οποιαδήποτε μορφή,,τι δεν υπάρχει στον κόσμο ΄που ζούμε,είναι ακριβώς ο ιδιαίτερος  τρόπος, με τον οποίο αντιλαμzzβανόμαστε τον κόσμο.(Εd:Régis 62)

I do not know if God  is a  mathematician.But mathematics is the loom upon which God weaves  the fabric of the Universe …The fact that reality can be  described or approximated by simple mathematical expressions suggests to me that nature has mathematics in its core.
Clifford Pickover


The back of a tiger could have been a blank canvas. Instead, nature painted the big cat with parallel stripes, evenly spaced and perpendicular to the spine. Scientists don't know exactly how stripes develop, but since the 1950s, mathematicians have been modeling possible scenarios. Now researchers assemble a range of these models into a single equation to identify what variables control stripe formation in living things.

This image shows simulations of Turing stripes. On the left, stripes are evenly spaced, but their direction is variable. On the right, a signaling gradient has made the stripes align in the same direction.
Credit: Tom Hiscock
The back of a tiger could have been a blank canvas. Instead, nature painted the big cat with parallel stripes, evenly spaced and perpendicular to the spine. Scientists don't know exactly how stripes develop, but since the 1950s, mathematicians have been modeling possible scenarios. In Cell Systems on December 23, Harvard researchers assemble a range of these models into a single equation to identify what variables control stripe formation in living things.

"We wanted a very simple model in hopes that it would be big picture enough to include all of these different explanations," says lead author Tom Hiscock, a PhD student in Sean Megason's systems biology lab at Harvard Medical School. "We now get to ask what is common among molecular, cellular, and mechanical hypotheses for how living things orient the directions of stripes, which can then tell you what kinds of experiments will (or won't) distinguish between them."

Stripes are surprisingly simple to model mathematically (and much of the early work on the subject was by Alan Turing of "The Imitation Game" fame). These patterns emerge when interacting substances create waves of high and low concentrations of, for example, a pigment, chemical, or type of cell. What Turing's model doesn't explain is how stripes orient themselves in one particular direction.

Hiscock's investigation focused on orientation--e.g., why tiger stripes are perpendicular to its body while zebrafish stripes are horizontal. One surprise from his integrated model is that it takes only a small change to the model to switch whether the stripes are vertical or horizontal. What we don't know is how this translates to living things--so, for a tiger, what is the variable that pushes the development of perpendicular stripes?

"We can describe what happens in stripe formation using this simple mathematical equation, but I don't think we know the nitty-gritty details of exactly what molecules or cells are mapping the formation of stripes," Hiscock says. Genetic mutants exist that can't form stripes or make spots instead, such as in zebrafish, but "the problem is you have a big network of interactions, and so any number of parameters can change the pattern," he adds.

His master model predicts three main perturbations that can affect how stripes orient: one is a change in "production gradient," which would be a substance that amplifies stripe pattern density; second is a change in "parameter gradient," a substance that changes one of the parameters involved in forming the stripe; and the last is a physical change in the direction of the molecular, cellular, or mechanical origin of the stripe.

Although this paper is based in theory, Hiscock believes that we are close to having the experimental tools that can decipher whether the math holds true in living systems.

Τα μαθηματικά αντικείμενα  συνίστανται από καθαρή σκέψη…. Η σχέση ανάμεσα στα μαθηματικά και την πραγματικότητα δεν είναι προφανής… H πραγματικότητα του χρόνου επηρεάζει σημαντικώς  τον ρόλο των μαθηματικών στη φυσική…Κάποιες πτυχές του πραγματικού σύμπαντος δεν θ’ αποδοθούν ποτέ με μαθηματικό τρόπο….η φύση δεν μπορεί να αιχμαλωτισθεί από κανένα λογικό ή μαθηματικό σύστημα…τα μαθηματικά δεν αντικατοπτρίζουν  όλες τις  ιδιότητες της φύσης…Τα μαθηματικά  δεν αντιστοιχούν στις πραγματικές φυσικές διαδικασίες,αλλά απλώς τις καταγράφουν…στην πραγματικότητητα τα μαθηματικά έπονται της φύσης…είναι ανόητη η άποψη ότι τα μαθηματικά προηγούνται της φύσης…Τα μαθηματικά είναι μια γλώσσα της επιστήμης…τα Μαθηματικά θα συνεχίσουν να είναι η θεραπαινίδα της επιστήμης,αλλά δεν μπορούν πλέον να είναι η Βασίλισσά τους…
Lee Smolin (“ O Χρόνος»)

Οι αλήθειες του μαθηματικού κόσμου υπάρχουν σε μια πραγματικότητα που δεν συλλαμβάνεται από κανένα σύστημα αξιωμάτων.
Roger Penrose

The back of a tiger could have been a blank canvas. Instead, nature painted the big cat with parallel stripes, evenly spaced and perpendicular to the spine. Scientists don't know exactly how stripes develop, but since the 1950s, mathematicians have been modeling possible scenarios. Now researchers assemble a range of these models into a single equation to identify what variables control stripe formation in living things.

This image shows simulations of Turing stripes. On the left, stripes are evenly spaced, but their direction is variable. On the right, a signaling gradient has made the stripes align in the same direction.
Credit: Tom Hiscock
The back of a tiger could have been a blank canvas. Instead, nature painted the big cat with parallel stripes, evenly spaced and perpendicular to the spine. Scientists don't know exactly how stripes develop, but since the 1950s, mathematicians have been modeling possible scenarios. In Cell Systems on December 23, Harvard researchers assemble a range of these models into a single equation to identify what variables control stripe formation in living things.

"We wanted a very simple model in hopes that it would be big picture enough to include all of these different explanations," says lead author Tom Hiscock, a PhD student in Sean Megason's systems biology lab at Harvard Medical School. "We now get to ask what is common among molecular, cellular, and mechanical hypotheses for how living things orient the directions of stripes, which can then tell you what kinds of experiments will (or won't) distinguish between them."

Stripes are surprisingly simple to model mathematically (and much of the early work on the subject was by Alan Turing of "The Imitation Game" fame). These patterns emerge when interacting substances create waves of high and low concentrations of, for example, a pigment, chemical, or type of cell. What Turing's model doesn't explain is how stripes orient themselves in one particular direction.

Hiscock's investigation focused on orientation--e.g., why tiger stripes are perpendicular to its body while zebrafish stripes are horizontal. One surprise from his integrated model is that it takes only a small change to the model to switch whether the stripes are vertical or horizontal. What we don't know is how this translates to living things--so, for a tiger, what is the variable that pushes the development of perpendicular stripes?

"We can describe what happens in stripe formation using this simple mathematical equation, but I don't think we know the nitty-gritty details of exactly what molecules or cells are mapping the formation of stripes," Hiscock says. Genetic mutants exist that can't form stripes or make spots instead, such as in zebrafish, but "the problem is you have a big network of interactions, and so any number of parameters can change the pattern," he adds.

His master model predicts three main perturbations that can affect how stripes orient: one is a change in "production gradient," which would be a substance that amplifies stripe pattern density; second is a change in "parameter gradient," a substance that changes one of the parameters involved in forming the stripe; and the last is a physical change in the direction of the molecular, cellular, or mechanical origin of the stripe.

Although this paper is based in theory, Hiscock believes that we are close to having the experimental tools that can decipher whether the math holds true in living systems.

Τα μαθηματικά αντικείμενα  συνίστανται από καθαρή σκέψη…. Η σχέση ανάμεσα στα μαθηματικά και την πραγματικότητα δεν είναι προφανής… H πραγματικότητα του χρόνου επηρεάζει σημαντικώς  τον ρόλο των μαθηματικών στη φυσική…Κάποιες πτυχές του πραγματικού σύμπαντος δεν θ’ αποδοθούν ποτέ με μαθηματικό τρόπο….η φύση δεν μπορεί να αιχμαλωτισθεί από κανένα λογικό ή μαθηματικό σύστημα…τα μαθηματικά δεν αντικατοπτρίζουν  όλες τις  ιδιότητες της φύσης…Τα μαθηματικά  δεν αντιστοιχούν στις πραγματικές φυσικές διαδικασίες,αλλά απλώς τις καταγράφουν…στην πραγματικότητητα τα μαθηματικά έπονται της φύσης…είναι ανόητη η άποψη ότι τα μαθηματικά προηγούνται της φύσης…Τα μαθηματικά είναι μια γλώσσα της επιστήμης…τα Μαθηματικά θα συνεχίσουν να είναι η θεραπαινίδα της επιστήμης,αλλά δεν μπορούν πλέον να είναι η Βασίλισσά τους…
Lee Smolin (“ O Χρόνος»)

Οι αλήθειες του μαθηματικού κόσμου υπάρχουν σε μια πραγματικότητα που δεν συλλαμβάνεται από κανένα σύστημα αξιωμάτων.
Roger Penrose

As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain they  do not refer to reality
Einstein
Αλλά
How is it possible that mathematics,a product of human thought that is independent of experience,fits so excellently the objects of physical reality?
Einstein

Μπέρτραν Ράσσελ., ο οποίος, αναφερόμενος; στα καθαρά μαθηματικά έλεγε ότι « αυτά είναι κάτι για τα οποία δεν ξέρουμε γιατί μιλάμε και ότι αυτό για το οποίο μιλάμε,δεν είναι αλήθεια»
-Ο Godel έλεγε ότι τα μαθηματικά, αντικείμενα υπάρχουν μακρυά σε μια άλλη διάσταση,ότι ασχολούνταν με την αντικειμενική ύπαρξη των μαθηματικών  αντικειμένων.,ότι ο χρόνος δεν υπάρχει υπό οποιαδήποτε μορφή,,τι δεν υπάρχει στον κόσμο ΄που ζούμε,είναι ακριβώς ο ιδιαίτερος  τρόπος, με τον οποίο αντιλαμzzβανόμαστε τον κόσμο.(Εd:Régis 62)

I do not know if God  is a  mathematician.But mathematics is the loom upon which God weaves  the fabric of the Universe …The fact that reality can be  described or approximated by simple mathematical expressions suggests to me that nature has mathematics in its core.

Clifford Pickover

There is no proof in science .The concept of proof belongs only to mathematics 

Ian Stevenson